No. Child support is not entered anywhere on a tax return and you are not the custodial parent. The custodial parent is the one who can claim the child as a dependent. If----and only if---you and the other parent have a signed Form 8332, you could enter the child on your return and get the child tax credit. Without a signed agreement, you do not enter anything at all about the child on your tax return.
Not if the other parent whom the child lives with is claiming them. Only enter the children that you are claiming on your tax return, if any.
The child is claimed by the parent who lives with the child. A child can only be claimed on one tax return.
No. Child support is not entered anywhere on a tax return and you are not the custodial parent. The custodial parent is the one who can claim the child as a dependent. If----and only if---you and the other parent have a signed Form 8332, you could enter the child on your return and get the child tax credit. Without a signed agreement, you do not enter anything at all about the child on your tax return.