Hi Tony,

I visited Israel after visiting the UAE, they let me into Israel without issues. Upon entering Israel I did indeed ask for my passport to not be stamped upon entry and exit and they were fine with that. They actually ask you whether or not you want your passport stamped (passport control).
LY on the other hand put me through hell and back trying to board the aircraft. Worst treatment I have ever had trying to board a plane. They did let me on the damn palne after a 3 + hour invasive security search by some very nasty peeps.
If you are trying to fly on US make sure coming back from TLV get back up passes on OAL. Our flight most of the times takes a weight penalty coming back out of TLV. Nonrevs have been known to have gotten stuck for days. If the light is wide open you should not have a problem just make sure you have a plan B. Avoid the summer, Jewish and Christian holidays, thats when flights are usually packed to the gills.