Assuming you qualify to pass on US Citizenship then you would file a CRBA to get the child a US passport.
Civil debts have nothing to do with Immigration.
Best bet is to head over to the Guides and then ask questions about anything you do not understand.
I assume the plan is for him to stay and adjust so he does not need a visa.
It is subjective but 6 months compared to 18 months plus would seem to quailfy
DCF seems to take around 3 months so I would use that as my timeline, of course like anything in life things can change
Did a Lawyer prepare the waiver?
There is no set time for a expedite application to be reviewed, last I saw they were taking about a month, and then of course the actual waiver would need to be reviewed.