New immigrants and qualifying returning residents receive a defined period during which foreign-sourced income is exempt from Israeli tax and, in many cases, from reporting. The exemption covers income arising abroad — not income generated by work performed in Israel, which is Israeli-sourced regardless of who pays it or where.
Generally not, if the work is performed in Israel. The source follows where the work is done, not where the employer or the payment sits.
The categories differ, and so do the conditions. Which one applies depends on how long you were previously non-resident.
In many cases it includes reporting relief, but not universally. This should be confirmed for your circumstances rather than assumed.