I didnt know that Qatar had normal relations with Israel. That is interesting...
Anyway, I wonder if there is any specific problems with being obviously Jewish with no Israeli stamp. Ie. you have a last name like Goldstein or something in any of the countries US citizens can travel to.
I personally know of a whole joint US/Israeli citizenship family that went to Dubai for a week and they were fine... Wonder if they just weren't noticed or if no one cared?