Your kids come first, and you should support them financially to the best of your ability with or without a court order telling you to do so. And I know you would.
Taking it up the azz legally is NOT "taking one for the team!" There's nothing that says you should financially enable your wife's affair. Why are you agreeing to an in-house separation? You've done nothing wrong, and you're not the one wanting out of the marriage, infidelity or not. If she wants "space," fine, let her go out and get her own place, and pay for it. The kids can split time between the two of your homes, in some agreed-upon arrangement.
Rules are controlling. A "rule" would be "You need to be home by 12 midnight." A "boundary" would be "I cannot live in a marriage where my wife stays out late at night and is having an affair." or "I cannot remain in a marriage where my children don't have access to their mother at 1 or 2 in the morning should they wake up sick or afraid and want you."
Does that make sense?
As for the finances, you need to separate those IMMEDIATELY (you should have already done so). Get her off of any joint accounts, and protect yourself. Why do you feel so powerless and like you have to pay for her waywardness? Am I missing something???