I was falsely accused of domestic violence and the baby momma put restraining order against me for no reason. Can I still fight for my son's custody?

Certainly.

There should be an evidentiary hearing coming up at which you will be given the opportunity to show your side of the story. By all means attend it and bring witnesses with whom you should try to prove that there was, as you say, no reason for the restraining order.

Yes you can fight for custody but it is a seperate issue. If you havent had a hearing yet on the restraining order then you need to attend that first before the order is made permanent.

Yes of course,

You need an attorney, your attorney will need to address the criminal charges against you, so you don't end up in jail.

You file for custody, (and also visitation until custody hearing is decided)
Do not get alone with the baby momma, always have someone with you, so there is no chance of new charges.