Adultery doesn’t commonly affect distribution of property or custody, unless there are extenuating circumstances (e.g. someone used marital assets to pay for a mistress or support a lover).
You may date while separated and before divorce, but sexual contact while technically married is still a crime in this state. (and you are married until the divorce is final). Although this may not affect ED or custody, by sleeping with a 3rd party before the divorce, you open that person up to being sued for Criminal Conversation and possibly Alienation of Affections (if you knew each other prior to separation). I’d advise against any sleepovers just because of appearances, even if nothing takes place.
(The other two questions I’m not quite so sure on. )