Ari's post step by step:
1.- Check-in to the flight to Russia: Use your Israeli Passport
2.- USA Immigration Exit Procedures: There are none
3.- Immigration entering Russia: Use your Israeli Passport
4.- Check-in to the flight to the US: Use your American passport
5.- Russia Immigration Exit Procedures: Use your Israeli Passport
6.- Immigration entering the US: Use your American passport

Based on your post, I guess you confuse check-in to your flight out of Russia with the Exit procedure. These are 2 completely different things and just use the passport you have to use. NEVER show 2 passports at an immigration booth, that's just calling for trouble...

Does anybody knows the answer to the last question? Do Americans get questioned when they enter their own country? In mine they just read your passport into the machine, stamp it and let you in...