Can a non-biological parent with shared parental responsibility and equal parental rights under UK law with his or her former partner (the biological parent) in a single sex relationship (gay or lesbian) take steps in Israel to prevent possible abuse of a UK child arrangements order allowing the biological parent to travel from the UK to Israel ,with their joint child, to visit his/her family there?

Yes! It is possible to apply for a Mirror Order in Israel of the UK child arrangements order so that it is recognized and enforceable under Israeli law.