I have seen it done, not easy, and in this situation adoption or SIJS are obvious routes.
I am going to assume the birth parents know that the child can't sponsor them on reaching 21.
If he was a Refugee he would have been handled by the UNHCR and judging from this post that seems highly unlikely.
As far as Asylum is concerned it used to be that you had to wait 5 months before you could apply for EAD due to this vary reason,
We have a whole forum dedicated to adjusting.
Once you file to adjust you are in a period of authorised stay. You certainly can get that in next week.
Overstays when adjusting through marriage to a USC are in any event forgiven.