Moslems

Yes, at the family court serving the area where you live. It will have jurisdiction to deal with the issue when the child is physically present in Israel and the need arises. The chief factor will be the child's welfare/good, as interpreted by Israeli civil law.

File for custody at the family court.

Yes, only at the family court.

Yes ! Parallel jurisdiction exists between the family (civil) and the Sha'ari court on the issues of child custody and child maintenance where both parents are Moslems and married to one another.

At the family court or the Sha'ari (Moslem Religious) court.

At the family court , the only court having jurisdiction over paternity. Paternity is not recognized outside of marriage, according to Islamic law. Paternity must be established before child maintenance can be set. If necessary, the mother can request that the court order genetic testing, to establish paternity, if the alleged father denies paternity. If he refuses, unjustifiably, this can play against him, and the court even has jurisdiction to declare him the father if his refusal to undergo genetic testing is unjustified, in its opinion.

The family court – what counts is the parties' religious affiliation when they got married.

The Sha'ari (Moslem religious) court according to Islamic religious law. What counts is the religious affiliation of the spouses at the time of their marriage. The divorce process will be a religious one, not a civil one.