Yes, when you are emancipated.
It's possible.
yes it is
no there is no way possible for that to happen
Yes you may challenge custody order.
It is possible for a non-US citizen to go to court in a custody case for a child. If one party is an illegal immigrant, they may be subject to deportation.
Anything is possible, but it's going to require a lot of documentation and very good argument on your behalf. One would have to prove to the court that sole rather than joint custody is in the best interest of the child.
You have to check the specifics of the custody/divorce agreements if there is one and the IRS tax code. It is possible that you would be entitlted to claim them as dependants. Consult a good tax attorney or CPA for the specifics.
It is possible that the other parent could petition the court to have your custody or visitation limited or revoked as being a bad influence on the child. What the court's ruling might be, no one can tell.
Is it possible in general? Yes. Whether it's possible in your specific case or not we can't say.
To do anything possible to get joint physical custody.
Yes it is possible for the mother to get custody, if she can prove that the child is better off with her. Custody can change at any time, regardless of where the parent live. Some judges might put limitations on somethings, but yes it is possible.
Yes, it is possible. You should visit or call the family court where the custody was ordered. You can obtain copies of any court orders as long as the case is still on file.