You have the right to name the child Bill Gates III if you wish - and Dad has the right to file in court to establish paternity, visitation and child support, and to petition the court and request kiddo's name be hyphenated to include his.

Putting his name on the birth certificate and/or using his last name does not confer him any rights - he'll still need to go to court to establish visitation etc.

You will need to establish paternity legally before you can get child support.