Let me break it down this way:
OP says:
" now he's telling me that he's going to bring me to court cause he wants full custody. Is he going to be able to get it?"
So that is the question and circumstances we are dealing with. The OP goes on to say there are some bad things about the boyfriend that should prevent him from getting custody.
Then I said:
"The court doesn't know any of these things about the dad, they have to be proven. Otherwise he's entitled to share custody."
So now are we on the same page?
A man brings a woman to court to get custody. He is entitled to get custody from the court, unless there are things about him which would make him not eligible.