You should be able to travel with conviction as indicated above but best to have an attorney review your case.

No this shouldn’t present a problem.

If the adoption is legal, the child is no longer legally your child.

Best to have an attorney review the asylum application prior to withdrawing.

After filing AOS, request emergency travel document based on information provided.

I agree with my colleague re: marriage.

It is $535 per applicant for the I130.

Yes they can depart immediately upon receiving GC.

Yes you should be able to depending on length of the travel.

Yes refile with correct fees to USCIS.