I suggest some of the answers are leaving out an important part of this. He is showing an interest in his son by coming over often, feeding him, changing his diaper, helping him sleep. Visiting once or twice a week even if he's not consistent is a point in his favor. Have you considered that he loves his child now even tho he didn't seem to be involved at first. Do you know that it's important for children/babies to have a bond with both parents? Please think about what is best for your baby. Let go of your justified anger while you focus on what is best for your baby. Parenting time can be worked out.
You've said nothing that indicates yours and his baby would not be safe if left alone with baby. Depression in and of itself means nothing about babies safety. Millions of depressed parents have successfully parented their children. He takes care of baby at your house which shows he's capable of caring for his son.
Parenting rights are NOT connected to child support. I urge you to talk to a lawyer and a mental health expert. It seems to me your situation is more about anger than learning how to co-parent together. It's very important for the two of you build a relationship together for the sake of your son. I suggest going to mediation to work out your relationship, even if in the end, you decide to not allow him in your life. He has the right to be in his son's life. Know that one way or another the both of you will be in each other's life for at least 18+ years. Based on your post, terminating his parental rights will not be possible.
I wonder if the angry words, such as quitting his job, were made when you were fighting. A mediator can help you keep your comments to each other civil while the two of you find common ground.
Of course he should pay child support. I suggest that you're telling him he has to pay in order to see his baby. Saying that is not only not helpful but is cruel.Talk with a lawyer to know the rights of each parent. Then talk together with a professional so that both of you can be logical in making decisions. The most important issue is what is best for baby.
By the way, he can go to court. He can ask the court for shared custody. That is why you need to work on making this work. You still need to have a court's decision giving you custody, outlining parenting time and a child support plan. Legal Aid can help you. Their fees are on a sliding scale.