If a child is born with no father listed on the birth certificate. Mother marries and child's last name is changed to mothers new name. Is husband responsible for child support if the marriage does not last?

If he legally adopts the child, then probably.

Yes

Changing of the last name does not establish support obligation without other legal actions like adoption for example.

If the husband has not legally adopted the child then he has no obligations or rights to that child what-so-ever.