I'm going to tell my boss to fire my homophobic coworker
How is your coworker being homophobic?
He joked about gay people several times
While it is not good to tell your boss to do something, it is not good for the workplace to have somebody homophobic there. Definitely your concerns are valid.
They don't really care. He says sexist things too.
It would be right to inform your boss of this coworker's behavior. They could harm the companies appearance and your boss should treat this matter seriously. Bring evidence and other witnesses to back you up.