Were there ever accounts of atheists or atheist communities in early civilization? Or even before that? When were the earliest atheists ever documented? And are there in theories regarding early atheism, or early religion at that?

In all of the historical works I've read on this, the first discussion of (quasi-)systematic 'atheism' begins with Greece, somewhere close to the middle of the 1st millennium BCE. Bottéro (1995), for example, writes that '"atheism" or even "agnosticism," even in the most moderate way, are concepts and elements of a problem that was entirely unknown in ancient Mesopotamia and, furthermore, generally everywhere in the early first millennium B.C.'

Even when we get further into the 1st millennium, I'm basically unaware of any non-Greek expressions of it. There are a couple of verses in the (Hebrew) Psalms, though, that are somewhat interesting -- for example, there's Psalm 14, which begins 'The fool has said in his heart, "There is no God."' However, it's hard to say much about what exactly this meant to the composer (also, FWIW, the dating of the Psalms is quite unclear). For more on the Psalmic evidence, cf. Crenshaw 2005.

I'm not 100% qualified to comment on it, but you might also want to look into the Cārvāka school of ancient India, and some of the related philosophical schools.

My knowledge of the Far East is nonexistent; so I'm really just going to focus on Greek stuff.

Also, I want to mention that there was a pretty popular post on here a few weeks ago, in response to the question How were Atheists treated by Greek / Romans?. It was even gilded several times; but in my opinion, much of it was very problematic. It relied on the non-existence of separate 'sacred' vs. 'secular' spheres, and an overemphasis on performative/ritualistic aspects. (A couple of buried responses to this were similarly critical: e.g. this and this.)

There wasn't a word about Diagoras or Critias, Prodicus, Anaxagoras, Protagoras, Euhemerus, Theodorus, or Epicurus.

Pieter W. van der Horst, in an article called "The First Atheist," quotes Critias -- who secured a claim to fame by his appearance in Plato's Charmides and Protagoras -- that

it was, I believe that a shrewd and clever-minded man invented for mortals fear of the gods, so that there might be a deterrent for the wicked, even if they act or say or think anything in secret. Hence from this source the divine was introduced [with the claim] that there is a deity who enjoys imperishable life, hearing and seeing with his mind, his thought and attention on all things, his nature so divine that he will hear whatever is said among mortals and be able to see whatever is done. If ever you plot some evil deed in silence, even this will not escape the gods, for they have knowledge . . . It was thus, I think, that someone first persuaded mortals to believe that there is a race of gods.

However, van der Horst writes that even though "it was in all probability Critias who first invented a theory of the origin of religion that was based on a cynical lust for power," it was Diagoras who was "the first outspoken atheist." Cf. also Prodicus, Democritus, and Euhemerus for others who sought to 'rationalize'/naturalize religion to various degrees. Really, these are going to be some of your best people for early "theories" of religion.

Further, in Protagoras' On the Gods, "the opening sentence already declares that one cannot know whether or not gods exist and, if gods did exist, of what nature they would be, because it is too difficult to find that out and life is too short for it (fr. 80B4 D-K)."

van der Horst continues

The poet Cinesias even seems to have founded a club of infidels who convened on a regular basis in order to ridicule the gods during their symposia, and several Greek authors from the decades around the year 400 make clear that they knew persons who cherish the conviction that gods do not exist at all (see, e.g., Plato, Laws X, 908b-e).

Although the former sounds polemical, the latter (Plato) is instructive. I'm being too lazy to translate the Greek myself, so I'll just use a (slightly older) translation. It's not perfect; but I suppose it'll do:

For while those who, though they utterly disbelieve in the existence of the gods, possess by nature a just character, both hate the evil and, because of their dislike of injustice, are incapable of being induced to commit unjust actions, and flee from unjust men and love the just, on the other hand, those who, besides holding that the world is empty of gods, are afflicted by incontinence in respect of pleasures and pains, and possess also powerful memories and sharp wits—though both these classes share alike in the disease of atheism [τὸ μὲν μὴ νομίζειν θεοὺς ἀμφοῖν ἂν ἐνυπάρχοι κοινὸν πάθος], yet in respect of the amount of ruin they bring on other people, the latter class would work more and the former less of evil. (908b-c)

The first century BCE Philodemus also has some really important stuff; and he even differentiates between agnostics and atheists (as well as a third category of 'functional' atheists).

One could produce a voluminous bibliography of academic works on this. (Though, again, a lot of this is going to be oriented toward Greek thought.) I'd certainly recommend the Cambridge Companion to Atheism, though, as one of the first things. Marek Winiarczyk has done some excellent work relevant to all this – he's produced probably the most comprehensive bibliography on atheism in antiquity (“Bibliographie zum antiken Atheismus,” Elenchos 10 (1989): 103–92). Also, see his recent The "Sacred History" of Euhemerus of Messene.

For Diagoras, I'm shamelessly copying a bibliography from elsewhere:

most recently M. Winiarczyk, Diagorae Melii et Theodori Cyrenaei reliquiae (Leipzig, 1981), to be read with his “Erganzungen zu Diagoras und Theodoros,” Philologus 133 (1989): 151–52; Bremmer, “Religious Secrets and Secrecy in Classical Greece,” in H. Kippenberg and G. Stroumsa (eds.), Secrecy and Concealment (Leiden, 1995), pp. 61–78; Parker, Athenian Religion, p. 208; Obbink, Philodemus: On Piety, pp. 525–26; and J. Hordern, “Philodemus and the Poems of Diagoras,” Zeitschrift fur Papyrologie und Epigraphik 136 (2001): 33–38.

Other works addressing various figures and issues include...

D.T. Runia, "Atheists in Aetius," Mnemosyne 49 (1996), 542-576
M. Lefkowitz, "Was Euripides an Atheist?," Studi italiani di filologia classica 5 (1987), 149-166; and “‘Impiety’ and ‘Atheism’ in Euripides’ Dramas,” Classical Quarterly 39 (1989), 70–82.
Dirk Obbink, “The Atheism of Epicurus,” GRBS 30 (1989), 187-223
F. Romer, “Atheism, Impiety and the Limos Melios in Aristophanes’ Birds,” American Journal of Philology 115 (1994), 351–65
M. Davies, "Sisyphus and the Invention of Religion," Bulletin of the Institute of Classical Studies 36 (1989), 16-32.
A. Henrichs, "Two Doxographical Notes: Democritus and Prodicus on Religion," HSCP 79 (1975), 93-124
C. Riedweg, "The 'Atheistic' Fragment from Euripides' Bellerophontes," Illinois Classical Studies 15 (1990), 39-53

/u/Azand had previously listed some works – and some of these get to the polis problem.