In general, this is explained when bringing up Occlumency in the first place. If he was good enough, he could readily agree to dropping his shields, then just keep them up. How would they tell? Again, this is assuming that he's good enough, but it does mean that they could never be certain. If they were motivated enough, they could Imperios to make sure he actually did it, but it is an Unforgivable...

1) I agree w/ JoshuaZ, Dumbledore might not want to further antagonize Harry w/o quite possibly gaining anything from it. It might be worth the risk, though.

2) That is quite possible, but I think that Snape's biggest hurdle in this is that he is still seriously underestimating Harry. Though the fact that Quirrell had to deduce that the girl had a Time Turner, and not just know, means that he possibly wouldn't know that there's a Slytherin w/ a Time Turner, or even necessarily that there are any issued at the moment. He could, through checking his student's schedules, but he might not know offhand.

3) I really wouldn't say an order of magnitude smarter than Dumbledore, he's just tuned his mind to that kind of thinking through decades of experience. Again, I agree w/ JoshuaZ in that Moody wouldn't be thinking of him as an option. He honestly might not even know the information that would be required for him to make that deduction, like Harry's predilection for Muggle artifacts.

edit: I forgot to say, I did not think of the animagus potion as evidence of Harry's innocence, but rather of Voldemort's guilt. Rather than making Harry less likely, it merely makes Voldemort more likely.