Paternity Testing

Yes especially if you go on any state aide. They will go after the father to reimburse.

Yes he can. It's his legal right.
He will have to pay for it, though.

Absolutely, it's his right. If he requests it he'll have to pay for it, and if his paternity is proven he then has access to parental rights like visitation, as well as a financial obligation to the child.

You can be sued for anything by anyone pretty much. That's why we have so many frivolous lawsuits. ;)
But that is the reality. You go to court and the court will hear his side (his petition and supporting info) and yours (your "defense" to his claim/request/petition) and decide if there is any legal basis and go from there. If he is just some guy that you never met before, and his request is baseless, the case will be dismissed. If there are any grounds upon which he might base his claim of paternity, then the court may issue an order requiring paternity testing.
My guess is that he is wanting to have visitation with his child. Visitation and support are two separate issues. I cannot stress that enough. Totally and completely independent of each other. And yes, he can petition the court to establish paternity (with a blood test if necessary) and award him visitation with the child (assuming paternity is established).

yes.

anyone that you had a physical relationship with around the time of conception can request one. Same is true for if the guy thinks you were cheating ...