Yes if you can obtain proof that the non-custodial parent has a source of income or file based on the same. If you have proof, it's open and shut. If you do not, it would be up to the judge to decide whether or not the non-custodial is lying if they testify to the contrary.
no
no
Generally, no.
NO!
Yes
yes
no
Yes. They are still the child's parent and responsible for supporting their child.
no
Yes, it can. Moving in with the other parent is grounds for "flipping" child support payments. However, this must be done by court order.
If the noncustodial parent tries to keep the child, the custodial parent can get the noncustodial parent charged with kidnapping and contempt of court both can be jail time for the noncustodial.
No. Your obligation to support your own child takes priority over any subsequent support you offer for another person's child. The court would not decrease your child support obligation for that reason.