Stability of New Relationship & Relocation: My ‘ex’ wants has applied to the Family Court to relocate to the States with our children. She has married an American recently. Will the court take into account that the relationship is new, and might not be stable?

It will certainly investigate the stability of the new relationship, and a professional appointed to make a report and recommendations may even meet with the mother's new husband, as well as her and the children, and yourself.
Relocation From Israel