Unfortunately, our sources for this phase of Athenian history are quite short and not contemporary, and there's a lot that simply isn't known. The events you're describing are known only from the account in Herodotos (5.66-78), written about 430 BC, and from a shorter version, apparently largely based on Herodotos, in the Constitution of the Athenians (Ath.Pol. 20-22) written by a student of Aristotle around 330 BC. I've written an account of the events in response to an earlier question - you can find it here.

As to your questions:

Do we know who the Athenians executed? Members of Isagoras aristocratic class?

Both sources refer to these men only as a group. Herodotos (5.72.2) calls them "the men of Isagoras' faction"; in the Ath.Pol., they are simply "friends of Isagoras" (20.3). The latter source adds that there were 300 of them, which Herodotos doesn't say.

As to who these men were, we don't know. They must have been men of prominent families if they desired to seize power and gained Spartan backing to do so. Herodotos suggests that Isagoras may have had close ties to the recently deposed dynasty of Peisistratid tyrants, and the men who supported him may have been worried about losing influence or even facing backlash now that the tyrants were gone. Their first act, the exile of 700 rival families including the very wealthy and influential Alkmaionids, also suggests fear of losing power again, or at least having to fight a protracted civil war to retain it.

The only named captive from the Akropolis, however, was Timesitheos of Delphi. Herodotos felt him worthy of record because "his achievements of strength and courage were quite formidable" (5.72.4).

Why Cleomenes I and Damaratos decide to fall back instead of moving on to Athens?

This was not so much their decision as a decision their allies made for them. At this point, according to Herodotos, it was clear that Kleomenes was using the army of the Peloponnesian League entirely to pursue his own interest, that is, supporting Isagoras' coup. He allegedly called up the army of the allies without telling them the target of the march (5.74.1). When they reached Eleusis, the Corinthians understood what they were being used to achieve, and declared it an unjust act they would have no share in. Demaratos apparently decided he agreed with them - probably because he was unwilling to help Kleomenes further cement his position of superior influence - and also led his men away from Athens. Kleomenes' army then fell apart, and he was forced to return to Sparta humiliated (5.75).

Who was leading the army of Boeotia against Athens?

This, we do not know. The Ath.Pol. doesn't mention the battle at all; our only source is Herodotos. It seems in this case he only talked to the Athenians about the events he described, because at no point in the narrative is he able to name a single Boiotian.