Even though I expect /u/iphikrates will come and make a post here, I'll give an answer because he tends to take a different approach to myself when answering questions and therefore provides different observations, which is a good thing. He is also better engaged than I am with Greek source material.
The term for phalanx as used today "transfers a Hellenistic and Roman meaning to a archaic and classical periods: the closely ordered, deep Macedonian...formation was called a phalanx as was a densely packed body of German or Gallic infantry." (Wheeler 192). The Greek phalanx consisted of a specific type of infantry, the hoplites, which is a term that did not come into use until the "late sixth and early fifth centuries" (192: citing Pind. Isthm 2.32, predating Xenophon). However, writers as 'late' as Thucydides (late 5th century BC) still did not use the term 'phalanx', but preferred taxis or parataxis. A phalanx is a unit of any type of troops regardless of the formation's shape, and denotes a battle-line as opposed to a marching column, but its definition as a "deep heavy infantry formation, usually in a square or rectangular shape" with multiple ranks is how the word is most often understood.
That said, 'phalanx' is a word generally used to describe an evolving tactical system that is peculiar to the Greeks. The linear formation and the importance of weight and mass that are typical of the phalanx are not peculiar to Greek warfare, and a "definite command structure and the use of column and line formations characterize state (as opposed to pre-state) warfare, and while we lack information on Near Eastern infantry formations in this period there is no reason to believe they did not utilise similar formations" (193). However, see this post here by /u/iphikrates for some distinctions between the Greek phalanx and phalanx-equivalents used by other contemporary societies.
Tracking the Greek development of the phalanx should be approached specific to the Greek experience and context, free from the assumption that it was only they who invented the phalanx, rather than heavily utilised a well-known and successful version of such a formation. It is clear that the traditional phalanx did not quite exist yet in the time of Homer (8th century BC: see 194-5). It has been suggested that changes in equipment afforded changes in formation. The hopla, the set of equipment from which hoplites derive their name, including the helmet, spear, sword and double-grip shield, the aspis, are evident in the 8th century, though not all of this equipment was necessary (195). Also important is that as the equipment an individual used was flexible and hoplites could actually be quite lightly equipped (perhaps only 12 kg) if they wished, the weight of their equipment does not explain the immovability of their formations. The use of the large round shield also does not necessarily mean the phalanx had to be adopted to be used properly: Roman legionaries used a large and heavy shield and they were quite mobile (196). There is a transition in offensive weapons from the 7th century onward, where the spear became more prominent, and after c. 625 the spear was the normal weapon depicted in Greek art. This might suggest a general preference for closer formations of combatants as opposed to a Homeric emphasis on individual combat skills. However, "such developments were not uniform through the Greek world and the extent of experimentation can only be surmised" (196).
A change in Greek warrior ideals is important to take into account. This can be observed in the 7th century BC in which soldiers are encouraged to fight for the polis, and flight becomes more disgraceful than it did in the Illiad: "the benefit of the group now takes precedence" (197).* However, adding depth to these linear spear formations "probably belongs to the sixth century: Greek armies in the Persian Wars are customarily deep. (197). Light infantry became less prominent in between the late 7th and early 5th centuries. However, this is not to suppose that hoplites were not more flexible than mere members of a phalanx, hence a trend over time of lightening their armour rather than making it heavier and rendering themselves more immobile. "The notion of hoplites helpless outside the phalanx is a myth." (199).
Wheeler summarises the issue as follows:
In sum, the phalanx did not appear everywhere in the Greek world and certainly not simultaneously even on the mainland: it was hardly an inevitable phenomenon. Hoplite equipment...did not instigate suddenly a tactical revolution. The evolution from fluid engagements of individuals and small bands to masses in line to a closely order mass in depth operation as a unit progressed over the course of the seventh and sixth centuries. Precise details are lost. Perhaps only among Sicilian tyrants...did forces of 'all arms' develop in any degree comparable to Persian armies. On the mainland, however...the agonal system** and the phalanx marginalised cavalry and light infantry in conflicts of limited aims within a closed system of poleis" (199, 202). Wheeler than goes on to state that Athenian imperialism and the Persian Wars would call this system into question and force the further development of Greek tactics (202).
As for the OP's questions, the fact that the phalanx requires "discipline, training and even ground" to work properly is partly true (the Macedonian phalanx was the one that was really vulnerable on broken ground, but correspondingly far more effective on flat ground). However, in wars between city-states or in the case of the Persian expeditions, an agonal conflict or a 'battle by consent' was a decisive way of deciding any issue, especially considering the militia-armies of the Greek polis were not particularly suited to taking cities or occupying territory and Persian armies in Greece really needed to force a decision quickly before their logistical situation caught up with them. This made pitched battle, or 'battle by consent', a favourable thing, as it always has been through history when states are seeking a decisive military decision. As such, both sides would generally agree to fight on appropriate ground so that they could use their arms properly: the Greeks would generally not agree to battle on highly disadvantageous ground, and if they do not agree to battle and cannot be forced to fight if they don't want to (they can always stay in their city), then a decision cannot be reached. So battle by consent is to some extent usually conducted on ground of both sides choosing, though every little advantage that can be gained from the terrain will be squeezed by a good general.***
No heavily-populated place on earth other than Eurasia is an endless expanse of open ground. The closest to such an area in Greece was the plains of Thessaly. And what were the Thessalians known for? Cavalry. Much like the 'horse-peoples' of Eurasia! Also note the prominence of cavalry warfare (and archery) in the open deserts of the East. The more broken the terrain, the better suited infantry are to it. Greece had enough open ground, even if it wasn't totally flat, to be well-suited to hoplite warfare. If it wasn't, it wouldn't have become such a prominent way of fighting: after all, the Greeks mostly fought each other.
Sorry, this answer ended up quite long. Basically: the phalanx was not necessarily a uniquely Greek phenomenon, but the 'Greek' phalanx was of course peculiar to them. It originated in great part as a result of their evolution from a pre-state society to a state society and the development of the poleis, which encouraged warriors to fight for a group, and as a group, rather than for themselves. This occurred over time and inconsistently in the Greek world, but the evolution occurred mainly over the 7th and 6th centuries BC. The first literary mention of its usage was in the late 6th and early 5th century BC, but as can be seen from writers such as Thucydides, the mere use of the word 'phalanx' does not necessarily mean the formation was or was not used: other words such as taxis or parataxis were appropriate. The Greek phalanx was flexible enough that a bit of rocky ground did not stop it from functioning: indeed, completely flat and open country was actually far more suited to cavalry than heavy infantry. The Greek phalanx was a more flexible formation than the Macedonian phalanx and was well-suited to the terrain of Greece (and really, so was the Macedonian phalanx, to judge by its later performance).
*-The argument has also been made that a "hoplite revolution" occured in which the prominence of the hoplite and their associated formation, the phalanx, in Greek society. The "hoplite revolution" was a product of peculiarly Greek political and social institutions, and the uniqueness of the Greek city-state was tied to their unique 'hoplite class'. This is just the broad strokes of this orthodox idea, and it would suggest that the Greek phalanx was also a unique product of Greece's unique social and political system. For more on this see Viggiano from Men of Bronze: Hoplite Warfare in Ancient Greece, 112-133.
** This is a reference to agon, which basically means in this context 'pitched battle', or 'battle by consent', in the context of the constant competition between Greek cities.
*** /u/iphikrates and others well-informed on Greek warfare might possibly argue that such battles by consent were not necessarily the norm, but they still occurred and hoplites were clearly well-suited to them, though they were no doubt well-suited to other styles of fighting too.
Source: Wheeler from the Cambridge History of Greek and Roman Warfare, 186-223
Per this comment, by /u/iphikrates , the phalanx is first mentioned by Xenophon
Might as well page him to this thread anyways.
In reply to your questions:
How did the phalanx originate?
I answered this here. The short version is that we don't know, because the period in which it happened is very poorly attested. It's most likely to have been a gradual development from earlier massed infantry crowds, occurring around the time the number of men equipped as hoplites increased dramatically (late 6th century BC).
When was the first mention of its usage?
I discussed this in the thread linked above, and also throughout this thread. The first mention of the word to indicate a single, unbroken, regular infantry formation is in the works of Xenophon in the early 4th century BC. However, there are earlier descriptions of formations we would call a phalanx; the earliest of these is a passage from Herodotos describing a scene set in 480 BC. It's plausible to suggest that technical terminology took some time to catch up to military reality.
Why would the Greeks adopt the phalanx?
Scholars have wondered about this for over a century: why would rugged Greece produce a type of fighting that requires open plains? But this question is predicated on the idea that open battle is the only way the Greeks fought. The account of the Persian Wars shows how wrong this is; the Greeks are constantly trying to hold bottlenecks, build fortifications, and refuse pitched battle in the open. What they adopted was a way to use masses of heavy infantry effectively in battle if they needed to, while always retaining the option of not letting it get to that point.
Of course, pitched battle in the plain certainly happened every now and then. To some extent, the choice of ground was the result of the size of Greek armies (where else are you going to put them?). But another important part of this is something u/PapiriusCursor hinted at in his post: no one wanted to be at a disadvantage. There are many examples of hoplite armies forming up on hills and awaiting battle, only to find that their enemies refused to attack. The Greeks were well aware of the danger of attacking an enemy in his chosen ground. If there was to be any kind of decision at all, it was therefore sometimes necessary for one or both sides to relinquish their advantages and meet each other in the open. This is not to say that they agreed on the place, though - more that they were willing to show a degree of vulnerability in order to lure the enemy into a fight. It also still left them with a range of ways to wrong-foot their opponents (exploiting time, remaining terrain features, reserves etc.), so that there is almost no battle in Greek history that may be said to have started on fair terms.
Also, the phalanx requires discipline, training
On the contrary, the hoplite phalanx (that is Greek, instead of Macedonian phalanx) requires little to no training. Which might be why it was adopted, an effective way to use untrained heavy infantry in battle. Only the Spartans are known to have trained, and even they did only basic formation drills. See here and here and many other threads like it all courtesy of /u/Iphikrates.