If a 16-year-old has a child with a 19-year-old can he sign the birth certificate so that the child would have his last name without getting in trouble with the law?

Depends on how old the 16 year old is. Typically 16 is the age of consent to sexual activity. There is probably not going to be any criminal charges. That isn't to say that the parents won't have something to say about the situation.