The short answer to your question is that, yes, there would have been real people with the names of heroes and gods, but no, they were not common.

The long answer is that Ancient Greek naming conventions are a complex and fascinating topic. On the one hand, there are no surviving sources that explain to us how or why the Greeks came up with names for their children. Ancient authors do not seem to have found this topic interesting enough to write about. On the other hand, Greek names form a tremendously large body of evidence - the Lexicon of Greek Personal Names claims to have published as many as 215,000 so far - and the fact that most names are composed of words with a direct meaning in Greek means that they present a unique window into the social world of Ancient Greece. Because of their components and their meaning, names can often tell us about people's place of origin, family ties, social status, cult practices, looks, and so on.

With the exception of some restrictions on the naming of slaves, there do not seem to have been any particular rules about what people could and could not be named. We might expect, for instance, that it was frowned upon to give your child the literal name of a god or goddess, but there's no evidence that the Greeks actually found it taboo. Admittedly the names of divinities, while becoming more popular after the 1st century AD, are rare enough in the Classical period (5th-4th centuries BC) that individual known cases have been extensively discussed by scholars. However, they are not altogether absent. There certainly were people named Artemis and Leto. This shows that there were no hard limitations to what a child could be named, but only conventions by which most names were chosen. The only names that genuinely don't seem to have been used at all were those of underworld gods (Persephone, Hades).

So what were the conventions they stuck to? One of the most powerful factors, especially among elite families, was the names of ancestors; some sons were named directly after their fathers (such as Perikles, son of the famous Perikles), while others were named after their grandfathers (like Kleisthenes, father of democracy, grandson of Kleisthenes the tyrant of Sikyon). Modern scholars are on pretty firm ground when they assume that people with similar names are related; often a name would "run" in just one family, sometimes for centuries. A related strong influence was the desire to express social status (again especially among the rich), which meant that many wealthy people would have names that contained words like aristo- (the best), -archos (leader), and especially mutations of the word hippos (horse). The latter is often regarded as a firm indicator of high status, since only the richest men in Greece would be able to afford to own horses, and horse-riding was the favourite pursuit of the leisure class. Famous examples include Perikles' father Xanthippos ("yellow horse"), the physician Hippokrates ("horse power"), and Philippos II of Macedon ("horse lover").

If there were no particular traditions binding new parents, they would be able to choose a name they liked. Endless possibilities are known. Particularly striking to us, though not necessarily the most common, are male and female names that contain a direct reference to warfare - Kallimachos ("beautiful battle"), Archestrate ("army-leader"), Nikomachos ("victory in battle"), Andromache ("battle of men"), Deinomache ("terrible in battle"). Other names are more straightforward, like Leon ("lion"), Kephalon ("head") or Melissa ("honey"). People were named after cities, mountains, and, very commonly, rivers; Aristotle complains about rare and ridiculous names like Hermokaikoxanthos, a pile-up of the names of 3 rivers. Generally, words like kalos (beauty), stratos (army), agora (marketplace), demos/damos (people), -anax (king) and old values like bia (strength) and kratos (power) occur often in what we might call "posh" Greek names. The combinations don't always make sense; what sort of a name is Isagoras ("equal marketplace") or Iphikrates ("strong strength")?

Interestingly, while the actual outright names of gods are extremely rare, by far the most common type of name in Ancient Greece was actually the indirect reference to a god or goddess. There were 2 ways to do this. First, the name of a god could be easily adapted into an adjective form, so as to become a name derived from a god rather than the god's name itself. Some of the most common were adaptations of Apollo (for example into Appolonios), Dionysos (Dionysios), Artemis (Artemisia) or Demeter (Demetrios); the latter is still in common use now, in its vowel-shifted form of Dimitri. One of the most famous examples of such a godlike name is Alexander the Great's companion Hephaistion (from Hephaistos). The second way to adapt divine names was to add words to them - most commonly kleos (glory),-doros or -dotos (gift, given). So we have authors like Diodoros ("god-given" - the reference is to Zeus), Apollodoros, Asklepiodotos; in the Roman era, when Eastern gods made their way into Greek lands, new names like Isidoros (gift of Isis) crop up. While it has proved impossible to say with categorical certainty that names are an indication of which gods were being worshipped in a given place, regional patterns are very clear in the evidence, and names derived from gods are at least a clear indication of which divinities were considered important. Perhaps the most touching are the few cases when parents who consulted the oracle on matters related to childbirth would name their child after the god who had advised them or after the place of the oracle.

Given all these factors and trends, it's perhaps easy to understand why the literal names of heroes were not commonly used as names, even if there were no strict rules or moral taboos against them. There wouldn't have been any family tradition to do so; there was no social credit in making pretentious references; since mythological figures were not always the recipients of cults, they wouldn't often have been credited for advice or protection in childbirth. It's possible that their names would have been regarded as old-fashioned or an ill omen, given the fates of most of them. According to the searchable part of the Lexicon, Ikaros is attested just 20 times; Narkissos a more respectable 74 times. I can't find a single person named Odysseus. Only the hero Iason has a famous "real-life" counterpart in the 4th-century Thessalian tyrant Iason of Pherai.

Source for most of this: R. Parker, 'Theophoric names and the history of Greek religion', in S. Hornblower/E. Matthews (eds.) Greek Personal Names: Their Value as Evidence (2000), 53-79

I have a (hopefully) up-to-snuff answer for this one.

A quick bit of background first: Ancient Greek names were commonly made up of two parts: a given name, and a patronym. In some cases a third part was added, denoting ethnicity or locality of birth. Given names were frequently dithematic (made up of two parts), for example "Socrates", from "sos" (σως), "unwounded or safe", and "kratos" (κρατος), "power". This is very common among Ancient Greek names. "Philippos", "lover of horses", etc. Patronyms were derived from the individual's father.

Names derived from gods and goddesses ("Theophoric", or "god-carrying" names)were fairly common. For example, derivations of "Dionysos" were common among both men and women. So too were names like "Apollodotos", "given by Apollo" (think, "god's gift" sort of thing). Gods associated with the underworld and death were not used, or at least not frequently (no "Hades", "Persephone", etc.), likely because they were thought to be cursed or unlucky.

Simple nouns were another popular source of names. These can run the range from basically descriptive to outright insulting-sounding. The word "aischros", "ugly", is the source of at least twenty known names, and is the basis for the name of the tragedian "Aischylos", better known in the Latin form as "Aeschylus". Parts of the body, traits, colors, etc. were all used in these simple names ("Kephalos", from the word for "head", "Euthyphro" of Platonic dialogue fame, from "εὐθῠ́ς", meaning "direct"). Animal-based names were also common ("Moschos", derived from the word for "heifer"). A personal favorite is Kopreus ("shitty"). The supposition is that a lot of these names are derived from affectionate nicknames for kids that continued on into adulthood and thus entered the historical record.

The names you are asking about, though, are heroic names (those associated with great heroes). "Hercules" (the Latin form of the Greek "Heracles", from the name of the goddess Hera, who despised him because he was the product of a union between her husband Zeus and a mortal woman), Icarus, Odysseus, etc. were not popular in Classical Greece. The etymology of some of these names is disputed or unknown, and some of them (Odysseus, for example, the derivation of which is disputed, but associated in scholarship with words meaning "to wail", "to be wrought against", etc.) have a decidedly dark turn. Not to say they were never used, but they weren't common in the Classical or Hellenic periods. Why is unclear. Perhaps it was considered too great a burden to name a child after someone who was so revered. However, after the Roman conquest of Greece a taste for Heroic naming arose and continued until Christianity became mainstream. At that point biblical names began to be more popular.

See:

Parker, 2000 (in the volume listed below): https://www.britac.ac.uk/pubs/proc/files/104p053.pdf

Hornblower and Matthews (eds.), 2000 (for a broader discussion of the topic): https://www.britac.ac.uk/pubs/proc/volumes/pba104.html

The absolute definitive work on this subject is the voluminous "Lexicon of Greek Personal Names", started by the Oxford classisist P.M. Fraser and still undergoing revision.

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There are a lot of layers in your question and spanning multiple eras, I will try to answer the best, but I'm not a historian, I will answer more from the linguistics involved, and the mods may remove the post if it doesn't fit the community guidelines.

There are thousands of personal Greek names recorded, but most of them come from the period after the 5th century BC, because this is the start of Greek Classical era we are most familiar with and Ancient Greeks had writing to document them down. Note that I said 'most of them'. Some of them are discovered more relatively recently, and they are older than the Greek Classical era, from Mycenaean times, which added to the inventory. But we have a lot of blanks in Greek history when there is no writing involved:

• Pre-Mycenaean Greeks (before c.1650 BC): Greeks had no writing, so we have no records on their names.
• Mycenaean Civilization (c.1650-1200 BC): Greeks seemingly conquered the Minoan civilization, and adopted many elements of their culture, from the palatial living, thalassocratic society, to even the writing style. Greek at this time began being written in Linear B, a syllabary, meaning like the system Japanese today it's written, with syllables. This was adopted from the Minoan writing system, Linear A, they use almost the same symbols (plus some differences), it's just that they are called Linear A and B respectively, because A comes first and B comes later. Syllabaries are an awful system to write Greek with, it just doesn't suit a language that has so many consonant clusters. Also, the two languages are different, Greek is Indo-European, Minoan doesn't seem to be related to any language we know of. However, this language might have left a large mark on Greek, a significant amount of Greek words cannot be traced back to Indo-European, these are called Pre-Greek and seem to include even some personal names. This is also the time Greeks come in contact with other civilizations, in particular Hittite, and some Greek personal names are recorded by them too. And they seem to have recorded non-Greek personal names too.
• Late Bronze Age Collapse (c.1200-1150 BC): The end of Mycenaean era, but also the time when Homeric stories were set, so partially relevant to your question. Linear B stopped being used.
• Greek Dark Ages (c.1100-800 BC): Greeks have no writing. Stories that would later be a staple of Greek literature are transmitted via bards.
• Archaic Greek Period and Classical Era (c.800-323 BC): I included these two distinct Greek periods in one because for your question they do not make much difference. This is the time when the Greek alphabet is developed, adapted from the Phoenician one, which is a much better match for the language than Linear B is, and they start writing down folktales and bard stories, often even making some changes to put several different stories in one single narrative, to try to make sense of them. For example, Athena had several stories who her father was, some said Zeus, other said Triton. Greeks of these period compromised and made Zeus her birth father, and Triton her foster father. This is not irrelevant to the question, by trying to make sense of myths or titles, Greeks created some personal names that never actually existed. For example Pleiades supposedly named for their mother Pleione, except we now understand Pleiades comes from πλεῖν meaning 'to sail' reflecting their importance in determining the sailing season in Mediterranean sea, Pleione, their mother, was created later to explain their name. The -des ending can either mean plural, or is used as "son(s) of", "daughter(s) of", Classical Greek scholars mistakenly assumed the latter meaning.

Despite the obvious difficulty Ancient Greeks created for us to make sense of, it's from the last period mentioned we know the most about Ancient Greek personal names, because of how much writing we had. The decipherment of Linear B added more to the inventory, but we still already had an idea on Greek names before that. Ancient Greek names usually are:

• Compound names: Inherited from Indo-European naming practices, for example Philippos, from φιλέω (philéō) "I love" and ἵππος (híppos) "horse". The name "lover of horses" might seem weird now, it was not unusual for a society that placed so much importance to them and was so reliant on them. Germanic society, for example, had Rosamund meaning 'defender of horses', from hros, meaning "horse", and mund, meaning "protection". Indo-Aryan Asvaghosa: "tamer of horses"; in Iranian Vishtaspa: "possessing horses untied (for racing)" etc. I'm certain I just implied imply only horse-themed names could make up compound names, which is not what I intended, I meant to say these names reflected what were important to the society, there were also names like Alexandros 'defender of men', Sophocles from σοφός (sophós, “clever”) +‎ -κλῆς (-klês, “glory, fame”) etc.
• Theophoric names: Names that included either θεός (theós, “god”) as part of their scheme or names of the Greek deities. For example, Theodoros means 'gift of god', while Demetrios comes from Demeter, the goddess of harvest and agriculture, itself literally meaning 'earth mother', however context matters, Demetrios is a male name (Demetria is feminine), it's not likely he was named 'earth mother', rather he was named after the goddess.

But we also have:

• Shortened names: Compound names and theophoric names had their shortened forms, just like names today have, Phil in English, for example.
• Single-meaning names: Names with a single meaning that probably originated as nicknames first, then that's the name they became known with. Ὅμηρος Homer means 'hostage'.
• Inherited names: Names that do not have a translation in Greek and are presumed to come from other languages Ancient Greeks came in contact with, usually taken to be Minoan, Luwian (the language Trojans presumably spoke, which is controversial in itself), Hittite, Phrygian etc. They might have also been the source of some Greek suffixes and prefixes.

I had to mention all this first before addressing your question:

• Hercules: is the Roman form of Greek Ἡρακλῆς, Hēraklēs, which translates into "Glory of Hera" making it a theophoric name, however his birth name was Ἀλκαῖος Alkaîos from ἀλκή (alkḗ, “strength”) +‎ suffix -αῖος (-aîos). He was renamed to please Hera. It's a plausible Greek name (both of them are), but we are not sure if it was used before Archaic Greek period, because we don't have sufficient data. Still, it is possible. The name however became more common later in the Roman Empire.
• Icarus: Roman form of Greek Ἴκαρος Ikaros, earlier *Ϝίκαρος Wikaros, a name with uncertain etymology. It most likely comes from another language, so not used among Greeks.
• Midas: A Phrygian king, and a non-Greek name as well. Perhaps it was common among Phrygians, but definitely not Greeks.
• Narcissus: Roman form of Greek Νάρκισσος, Nárkissos. It's not a Greek name either, especially not with that -issos suffix. The double -ss-, according to Beekes, was perhaps originally pronounced something like *ts which is not native to Greek phonology. The name is used for a flower, and Pliny the Elder said that the plant was named for its fragrance (ναρκάω narkao, "I grow numb").
• Odysseus: has also a Roman form Ulysses, and other forms where you see the d/l variation, which is important because it points at a pre-Greek origin (usually taken to be Minoan). The l/d variation occurs in words of pre-Greek origin, for example labyrinthos/dapurintos 'labyrinth, maze' < presumably from labrys 'double-sided axe', an important symbol found in Minoan frescoes (how it came to mean 'labyrinth' is a question in itself, it is hotly debated). It also has -ss- which as explained in the entry for Narcissus, it was probably pronounced *ts, a phoneme not native to Greek, that Greeks tried to approximate. That said, ancient authors have connected his name with odussomai (ὀδύσσομαι) “to be wroth against, to hate”, which follows the same theme-naming as Achilles, itself a compound name made up of ἄχος (ákhos) "distress, pain, sorrow, grief" + λαός (laós) "people, soldiers, nation", and while not common in Classical era, it did exist in Mycenaean era, it was recorded twice in Linear B as a-ki-re-u and a-ki-re-we (dative form). Menelaos follows the same theme-naming, from μένος "vigor, rage, power" + λαός "people," "wrath of the people", though the name itself is not recorded in Linear B tablets, it is however, plausible.

We do have three legendary names recorded in not Greek, but Hittite tablets:

• Attarsiya: presumed to be the Hittite form of Atreus, father of Agamemnon and Menelaus, from ἀ-, "no" + τρέω, "tremble", meaning "fearless"
• Akagamunas: presumed to be the Hittite form of Agamemnon, from ἄγαν, "very much" + μέδομαι, "think on", meaning something like "very steadfast"
• Tawagalawa: presumed to be the Hittite form of Eteocles (back when it used to be pronounced *Etewoklewes), meaning "truly glorious", note the -kles in Herakles too.

TL,DR: Some of them yes, some of them were plausible even though not recorded, some of them were non-Greek names so unlikely to have been used by Greeks, some were created later to explain unclear etymologies to Classical Greeks, but were unlikely to be used in real life.

Sources

1. Naming Practices, Lexicon of Greek personal names, Oxford University
2. Eric H. Cline, "1177 B.C.: The Year Civilization Collapsed"
3. R. S. P . Beekes, "Pre-Greek, a language reconstructed"
4. R. S. P . Beekes, "Pre-Greek: Phonology, Morphology, Lexicon",

Follow-up question: what were the generalized naming conventions in the region at that time period?