If I'm not the child's father and at the time of birth I do register him on the birth certificate, do I automatically obtain Parental responsibility?

Depends on where you live. In some states, signing the birth certificate is equivalent to an acknowledgement of paternity. In other states it may be considered perjury because you knowingly falsified a legal document.

Even if you are deemed to be the legal father, if the bio father decides to challenge it, he usually can.

Maybe, if you explain the full circumstances we can help more.