As compared to what? Is the mother a child abuser? It depends on the circumstances.
Yes, he is still the father. It's just that the child is not living with him.
yes, and file for custody based on custodial interference.
Can, but the father can challenge it in probate.
As unfortunate as this may be it is possible for the father to get custody of the baby, but not full custody unless the mother is proven unfit. Because reguardless of what happened between the couple the father is still the father of the child, and the father still has his rights as a father to be a part of his child's life.
You have the visitation rights that were established in the divorce, and you have no custody rights.
He can still gain custody as the presumptive father.
yes
You would need approval to return to the country to file a challenge.
Yes, if the father is not given custody he will be obligated to keep paying support to whomever the court awards custody or guardianship of the children.
The mother can still have sole legal and physical custody when the father is awarded visitations. Custody and visitations are separate matters. The mother would be required to obey the visitation schedule.
No, if they are married they have equal custody.
Yes, this is possible.