One
The father signed birth certificate at birth is he financially responsible at that time or does child support order have to be put into place?

Two
The father quit his $1000. A week job and says he cannot afford to help me pay for formula diapers or clothing for his child. Can I file abandonment with proof in text that he will not help me pay for anything for him because he doesn't have a job now?

Child support has to ordered by a court. Child abandonment would be used as grounds for simething else like your husband adopting the child. For any better answer than that we need your general area like what state you are in because local laws very.

Even if he signed the birth certificate he may not be liable for child support unless it is proven he's the bio father or he is designated the legal father by a court.

You don't usually "file" for abandonment (except in a very few areas). In the situation you describe, you file, in court, for child support. Support will be awarded based on income. If he is willing to give up his career to not pay support, he can get away with it.

He does not have to, just pay for supplies, he should be paying child support based on his income. Does not matter he quit a job. So file for court ordered child support and custody of child