If your child has a Thai birth certificate and your named on it. You could get an visa extension based on supporting them. There is one benefit that is you only need to have 400KTHB on the date of the application, rather than 2 months before for a Thai wife, even so your still need marriage certificate etc. Also check with where be living some immigration offices have their own requirements
I think its a "residence certifate" I have ti get my father in law to fill mine in every year, plus a copy of his id. I don't pay rent. I don't think its a rental agreement. Try and translate the title with Goole translate.