How often should a player be able to backstab or ambush an enemy in a fight?
Elven Rangers for example can vanish in the underbrush on a 4+, so does that mean that my monster has to succeed on a listen check to prevent getting double damage on the next attack or is it impossible to hide once combat started?
Backstabs in mist old school games are kind of a one shot. The target must be surprised or severely disadvantaged (as outlined in specific combat maneuvers) to pull off. Once they know you are there they are moving around and trying to defend themselves too much for effective sneak attacks.
I generally allow a backstab again if the thief is able to break line of sight (often through invisibility or an elven cloak), move silently up to the target, and the target is distracted. But it's definitely a judgment call.
You don't want to make it so difficult to get a backstab that the thief's core power is useless, but you also don't want it to be so simple as 3.5's "flanking the target". The thief isn't meant to be a "striker" in the 3.5 sense.
would you consider being engaged in combat a sufficient distraction? IE: the rest of the party has formed up into ranks and is fighting while the theif/assassin sneaks around the back to stab a mage?
It would quite depend. If the room was dark, or there were obstacles blocking line of sight, and the fight was loud and noisy, then I'd permit it. On the other hand, in a brightly lit room, with the fighting confined to a low hum, I'd probably not.
as someone who has played probably 95% 3.5 and later… this non-tactical stuff is going to take some time for me to get used to 
If you like t actical combat, you can play ACKS in a more tactical manner -- lay out miniatures with 5' block, use the engagement rules as a zone-of-control, and so on. It works quite well. One of our local GMs prefers to play that way.
If that's the route you go, then I think you could rule that Hide/Move is possible so long as the NPC target doesn't get an unobstructed line of sight to the sneaking character.
If we were playing in meatspace I might consider that, but because we are doing play-by-post, I have a vested interest in keeping it simple and sticking to more abstract combat depictions.
One more question: cleaving. Could an assassin hypothetically sneak up on some low-level ish guards and backstab all of them (provided enough damage was dealt) using the cleaving rules?
Yes. If none of the guards were aware of the assassin when he began attacking, or all of the guards were surprised, then any cleaves would also constitute backstabs.
Ouch!