Were the parents ever married? If not, was paternity ever established, and is there a court order regarding custody, visitation and support? If they were married, are they now divorced? In what state was any divorce judgment or order entered?
Yes, they were married and now divorced, the court order is not very clear to me, unfortunately, even after 7 years, they fight over that sometimes. Divorce was finalized in California.
As you've described the living situation, that cannot be true. They might have joint (not "joined") legal custody (which relates to decisions regarding schooling, religion, medical care, etc.), but they clearly don't have joint physical custody.
Oh duh, thank you for correcting me... and I'm sure you're right about the type of custody..
Can they "come after" you? I don't really know what that means, but the answer is probably yes. If you're asking about legal liability, that's fact-specific and can't really be answered in the abstract.
I understand and thank you for making that clear.
If you agree to allow their child to live with you, you would not acquire more rights as a result. You would be undertaking responsibilities and would owe legal duties to both the child and his parents. The parents could (and probably should) give you certain authority to act in their place (either by signing a power of attorney or by seeking a guardianship through the court).