Yes they can apply - there will be tough questions though.
I believe that you're guaranteed a citizenship when you marry a citizen, but being an illegal complicates the matter.
It is legal in the United States for an illegal immigrant to marry a US citizen. Once the marriage is legal, the illegal immigrant becomes a legal immigrant and can stay in the US for the course of the marriage.
It's not illegal.
No he cannot
If you are married, you can apply for divorce.
no not unless they where married to a USA partner.
Not unless they themselves are registered in court as an American citizen.
the US citizen shouldn't
=An immigrant is NOT an alien.==The immigrant would still be legal because they were married to a US citizen.=
Yes, an illegal immigrant from Bulgaria can get married in the United States. An American can choose to marry whomever they want.
If an undocumented (illegal) immigrant is married to a U.S. citizen, then they are a U.S. citizen. So their divorce from another U.S. citizen won't affect the status of their child at all.
Yes they retain citizenship.