I am a mother living in Israel. I am not married to the child's father, who is registered on her birth certificate. I don't live with him either. Can I move abroad with my child without him knowing or agreeing ?

No! Under Israeli law a biological father is a child's natural guardian even if unmarried or separated from the mother, and has clear rights. If you relocate abroad without the father's knowledge and consent, this amounts to an act of chid abduction,and you are liable to face Hague Convention proceedings.