Your girlfriend's son with another man is neither "our son" nor your step-son.
It's not clear what "all of this" might mean.
It sounds as though there is a court order regarding custody, visitation and support. Correct? However, it also sounds like the father has no involvement with the child (i.e., doesn't exercise his visitation rights) and doesn't not pay child support. Also correct?
Like it or not, this is the guy your girlfriend chose to have a child with, and he has parental rights until and unless a court says otherwise. Under the circumstances, the only way to get his rights terminated would be if you step up and marry the mother and seek to adopt the child. If dad remains a non-entity in the child's life, then you might be able to get dad's rights terminated so that you can adopt. Note that getting married isn't an absolute necessity, but judges may be disinclined to allow an adoption unless you do that.
I suggest you and your girlfriend consult with a local family law attorney for advice.