Melbourne, FL Immigration Law Firms & Lawyers

Problem with the case is as you say, the acquisition of US citizenship upon the birth of your child abroad would depend (usually) on establishing the parent's physical presence in the US for a period of time before the child's birth. And without the cooperation of the parent that can be difficult. The answer to your question is that you do not need the parent's cooperation. However, finding the documentation which would establish the physical presence can be problematic without that cooperation. I know that this answer places you between a rock and a hard place, but these cases require evidence and without the evidence to support the claim of acquisition of US citizenship, the claim will often fail. Wish I could give you a better answer. Perhaps hiring a private investigator in the US to find available records may be helpful, but this will have a cost and of course no promise that the investigative services will lead to anything. Good luck.

If she entered the country legally, and has overstayed her visa, the fact that she was working illegally will be overlooked if she later marries an American citizen who applies to adjust her status. 

As a general rule, USCIS wants for the marriage to predate the grant of your asylum status. So if this is the case then you'll have first adjust status, then naturalize in 5 years, and then adjust your wife.