There's no issue about your parental rights, Shamir. You're the father and you can prove it. She can put whoever she want on the birth documents, that doesn't change anything.
Assuming that you and mother can't work anything out, you should be getting ready to go to court as soon after the birth as you believe is necessary. But you need to think about what you want, and about the strategic implications. If the mother doesn't even put your name on the birth documents, and you have a history with her boyfriend, then it's pretty clear that you two can't get along for whatever reason. This means that getting shared custody is going to be an uphill battle (unless she agrees, and what are the odds of that?).
Similarly, I've seen a lot of fathers in your situation say that you want the child half the time, what's called "shared custody". But the same logic applies, and is in fact worse for you. I know you don't want to take the child away from the mother, but if you have a smoking gun then that's frankly easier than to get shared custody or joint custody over her objections.
So you'll need a smoking gun. Something concrete about her mental issues, or criminal activity, or documents drug/alcohol use, or being intransient. Something that would get the CAS on board and have them support you. If you don't have that, it's tough to get an infant away from the mother.
So if you're going to swing for fence here, you'll need that information from reliable third parties, and have a good plan of care yourself, what your routine for the child would be and who is helping you.
You won't be able to get an order that boyfriend not be around. I appreciate your reasons, but it's not going to happen in reality. If you have the evidence that he poses a risk of harm to the child, then you're better off seeking custody than an order for the boyfriend to stay away. I hope you can get a family lawyer on board who is good with such matters, and start getting a plan together. You won't get far doing an affidavit saying "boyfriend is a jerk", that isn't direct evidence as to why the child is better off with you than with mother.
Does that make sense? Please reply with question or comment, I'll be here.