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.
no
No there is no need for the child to go to Texas.
no
Generally, no.
Yes
no
yes
Yes. They are still the child's parent and responsible for supporting their child.
no
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.