Mother was married before but before her divorce she had been pregnant 2 times and since the mother was married. The fathers name did not get put on the birth certificate what happens next? If the mother wants child support from the father. Or does she have to prove who she slept with? Then, what happens to the kids If that guy is taking care of another man child?

Depends on the time frames. If the husband singned the birth certificate he will usually be deemed the legal father. For another man to be awarded paternity he would have to file a petition in Family Court and have a paternity test done.