A lot of ancient mediterranean cultures would have some sort of qualifier added to a gods name, although I can not think of any with only the names of the god. For example, Hannibal translates to Grace of Baal. Common qualifiers include doros (as in Apollodoros, Theodoros, etc) mean "Gift of" Apollo/ God etc, And klese ( as in Diokles, Hera-kles latinized to Hercules) means "praise be/to".

Derivatives of god names is also popular. examples include the popular Greek name Dionysios derived from Dionysos, and the Roman name Marcus probably relating to Mars.

Some Hebrew names surviving until today are references to God. Michael when translated from its original hebrew means "he who is like God"

Edit: not great at Hebrew, looks like Michael might more closely man "who is he like god?" Although my point of Hebrew names making references to God stands.

The primary phrase you are looking for is what are called theophoric names, which are essentially names that specifically invoke a deity. The prevalence of these varies widely by culture: to give an example from two generally rather related ones, Greek has a wide repertoire of theophoric names (Apollodorus=gift of Apollo, Hermokrates=strength of Hermes, Diokles=glory of Zeus, etc) while classical Latin, at least from a quick glance through my memory banks, has none. There can also be a great deal of social differentiation regarding naming within a culture. Egypt gives a good example, as quite literally every pharaoh has a theophoric name, often with some specific significance, such as Mentuhotep (Mentu is pleased) or Tutankhamen (the living image of Amun), while this was not necessarily the case among commoners, who might have names like "Tiye".

Now within the category of theophoric names there were some that were just the deity's name. Dionysius (derived from Dionysus) is a fairly common name, for example. They are somewhat rarer and I have never seen a name like "Zeus" but I think that is what you are looking for.

I have never seen a convincing explanation for cultural differences in theophoric naming practices (admittedly, this may be due to a lack of looking). It is something that lends itself well to people putting their conclusions before their argument--I have seen one statement, for example, that theophoric names were common in Phoenician naming, unlike the Greeks--which is just not correct.

Just a clarification- Muhammad is not a god's name, it's a prophet's name. Many people in the Christian world, including myself, are named after prophets. The name Jesus is exclusive to Iberia due to the historical power of Islam there, where Catholic Spaniards would name their sons Jesus as an equivalent to the many Muhammads in Iberia (which is kind of a false equivalency, but it stuck). Furthermore, naming someone Muhammad in many Islamic countries has far more to do with tradition rather than the name itself, so it's more of an honorary prefix. It's not uncommon for all the men in a family to be named Muhammad, and they would be distinguished by secondary names, for example.

Almost. Theophoric names, which included the deity's name as one component, were very common in the ancient Near East. To address your specific examples, several characters in the Old Testament have names that contain "Baal" as a theophoric component (and some were even Israelites, which attests to the origins of these stories in a time when polytheism was the norm). There's Phoenician king Ethbaal, Saul's son Ishbaal ("Man of Baal"), Jonathan's son Meribaal, and so on. Sometimes, other biblical passages will euphemize such names by replacing "baal" with "bosheth" ("shame"). Nebuchadnezzar contains the name of the Babylonian god "Nebo". Any Bible name ending in -iah contains "Yahweh" in it, and any name ending in -el contains "El" ("God"). "Jesus" itself is the Latinized version of "Joshua", a theophoric name.

It was quite popular to name children derivations of a god's name in the viking age. But not directly after a god. Amongst others, Þórðr, Þórstæinn and Þórbiôrn were popular derivations of the god Þórr (Thor). It was most likely not until much later, after the christianization of scandinavia, it was deemed acceptable to name children "Thor" or "Odin" directly.

I can only answer this in regards to Ancient Greece and Rome. The short answer is 'no'. There are some exceptions though.

I have never seen or heard of a person in Latin text directly using the name of a god as a proper given name. However, there are some individuals who are named in honor of a given god. Apollonius or Apollinaris are good examples of names that are in honor of gods (in this case Apollo.) These were often family or tribe names but did appear as first names as well.

The difference between the name of the god in each case would be the same as the difference of Christ and Christian in English if that helps your understanding.

Hope this helps address at least one facet of your question.

Wouldn't, in the OPs example, the kids be named after prophets of a given god, as opposed to the god itself. Like were there influential priests of Neptune or Baal or Set whose names lived on after their deaths?

A follow up: are there a lot of modern names derived from the names of old gods? Maybe even a lot that aren't obvious? I know there's a lot of modern names from Norse Mythology, but what about any of the other European mythologies?