Most court orders will require you to notify the court before moving out of the county or neighboring counties, to obtain the court's permission. Most orders also will require both parents' signatures to be on a passport application. Taking these two things together, unless your court order says otherwise, you will need to get court's approval to move out of the country with the child.
If you are on the other end of this, as the parent who is being left behind, you MUST file an order immediately, and preferably before the move takes place, or it will be very difficult to get the child back.
lost the child to whom?
her choice
No, you still need his permission. The court order says joint custody and if you take the child out of the country you make that impossible to work and therefor break the court order. That last part just means that wherever you go, you still have custody. Otherwise you could go on vacation abroad and lose custody just because you are out of the country even though you have his written permission to take the child with you. Your ex of course also needs your written permission if taking his child out of state or country.
With regard to child custody, the court must consider what is in the best interest of the child. A felony might hurt a parent's character, but it isn't necessarily a bar to having custody. It just depends on the specifics.
I'm assuming you are the father and a fit parent and then yes, you would probably get custody.
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.
You would need approval to return to the country to file a challenge.
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.
The custody issue would probably depend on the country in which it is tried and where the child resides. It is likely that the child will remain in the country he or she is living in.
Read the Uniform Child Custody Jurisdiction And Enforcement Act (UCCJEA). Most states have adopted it.
Though not specifically applicable to the UK, the arguments in this regard are important. See link
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.