You can give your child literally whatever name you want. Literally. You could give your child the last name clooney if you wanted. They can't stop you, and It doesn't affect whose dna he has.
Yes u can like she said when y'all go to court the judge will send him downstairs or upstairs depending on ur court lay out and he would do a DNA test there and y'all will have to come back to court
Yes you can but if he isnt on the birth certificate, they will need to do a DNA test before obtaining support for the child.