now, i am no eggspurt, but i dont like the plan. it takes aboot 1/2 hoor to walk thru the museum to get to the sistine and i am not so sure once you get there that you can reverse, even if you can you will add at least an hoor to your visit. also, do you have a st peters access? you can access SP directly from sistine via the hidden door, though you miss the famous staircase.....my .02.
literally a zillion restaurants, but rather than recommend, i can say avoid eating on Main drags, near tourist spots and at places with tourist or prix fixe menus.
our favorites are il giardino romano, la fraschetta di castel san angelo, da enzo 39 and that spaghetteria near the trevi...i need to find the name.
spaghetteria l'archetto-grumpy, a bit rude, credit machine randomly broken, but the menu is crazy with over 100 dishes and finger foods.
tip-be careful if you want to eat outside, smoking allowed outside and you can wind up feeling like you are in an ash tray.