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Only with the other parent's and the courts permission.
A change of custody must be filed through the courts. A change in child support should also be filed through the courts.
The mother has sole custody and control in all US States. see links below
lost the child to whom?
No, however the chances of you getting primary custody are slim to none. The other parent would have to be deceased, or proven unfit for the child. Which with 5 domestic violence charges, you would be proven unfit immediatley.
her choice
This is done through the courts.
Depends on the type of custody you are assigned in the courts. If the child is primarily at the other parent's home, then your share remains the same. Remember it all is to benefit the child. Many do resent the other parent having the funds but everyone needs to be realistic about the ongoing costs of raising a child. Costs of providing a home continue through the weekend. In my experience, custody and child support change over time as the child grows older and lives change. The better the parents can communicate, the better for all.
If you are not married and there is no custody or visitation order, she has custody automatically. The father have to prove paternity in court by a DNA test and then petition for custody or visitation. He can then also pay child support.
If you have sole custody no. However, can you still provide for the child? School etc? If she wants to prevent it then let her go to court.
Read the Uniform Child Custody Jurisdiction And Enforcement Act (UCCJEA). Most states have adopted it.
If the arrangement is with the consent of the custodial parent and will be permanent then the custody and child support orders must be modified to reflect the change in legal custody. The parent in Texas needs to have their custody formalized by a court order so they can enroll the child in school, consent to medical treatment, etc. If the child support order is not modified the non-custodial parent may be subject to the accumulation of child support arrears.