Yes, of course! In this case, the main focus is on the fact that the person is here without status. There may or may not be a way to fix the situation in immigration court, so a consultation with an immigration attorney is best.
Typically, having no status isn't a bar to matters such as marriage, divorce, and adoption. I guess a court could, however, raise an issue over how he can support the children if he's undocumented. Consult with a Pennsylvania family law attorney to see if an adoption case can be filed.
Yes. He has to convince USCIS that the marriage was entered into in good faith. Whether he can do that or not depends on a variety of factors, including the quality of joint documents that he produces.
It's time to get an immigration attorney involved if you don't already have one. USCIS could deny the application due to misrepresentation or require you to file a waiver. Being unaware isn't usually an excuse that they accept.