In the so-called "Nunnery scene,” do you think that Hamlet knows that he's being watched? Can you find any evidence of his awareness?

As is often the case in Hamlet, one can never know for sure what Hamlet is aware of or what he's thinking. My interpretation is that he probably does realize that Ophelia has been sent to him so that her father and Claudius can observe his actions and get some clue to whatever it is they fear Hamlet intends to do. This, of course, does not excuse or explain the cruelty of his behavior toward her, nor does it answer the question of how much of Hamlet's "madness" is real and how much is feigned. Part of the greatness of the play is rooted in this very ambiguity, in the openness to interpretation of Hamlet's mental state and his feelings towards Ophelia.
Most commentators have seemed to think that whatever the reality of his personal anguish, Hamlet is laying it on thick to make Ophelia believe he is insane. It can be argued that the following lines do indicate he also knows that the King and Polonius are eavesdropping:

I say we will have no more marriages. Those that are already married, all but one, shall live. The rest shall keep as they are.

The "one" probably means Claudius, whom Hamlet intends to kill. In my view, this is a threat he intends Claudius to hear. Earlier in the tirade, Hamlet already appears suspicious that Polonius may be listening in, when he asks Ophelia where her father is. When she says he's at home, his response is:

Let the doors be shut upon him, that he may play the fool nowhere but in's own house.

There is no special reason Hamlet should believe Ophelia that Polonius is at home, and the insulting remark he makes is probably one intended to be heard by him. Hamlet obviously thinks Polonius is a fool, though to this point he hasn't said so in such direct words. The statement may also be a warning to Polonius to stay out of the conflict between Hamlet and the King—that it is none of Polonius's business.
That said, there is no explicit proof that Hamlet knows Claudius and Polonius are listening. My view is only that it seems probable. One could also argue that he would not have poured his soul out in the "To be or not to be" soliloquy preceding this if he had any suspicions people were eavesdropping. But as stated, the fact of ambiguity and multiple interpretations is one of Shakespeare's strengths.