When does a PDA halt?
The setup places $aa\$$ on the stack and puts the machine in state $x$. We then have this sequence of moves on input $aa$:


Consume the first $a$ and pop the stack. We're then still in state $x$ with stack contents $a\$$. No other move is possible.
Consume the next $a$ and pop the stack. We're then still in state $x$ with stack contents $\$$. No other move is possible.
Seeing a $\$$ on the stack, the only possible move is to pop the $\$$ and pass to state $y$.
In state $y$ with no further input, there are no possible moves, so the machine halts, in a non-accepting state.


Notice that there's no nondeterminism here: perhaps you were mistaken in your first bullet point: indeed there is a $\$$ in the stack at move (1), but it's not on the top of the stack.

The deeper question is exactly what model of a PDA is being used here: some people allow multiple accept states and say that the PDA accepts a string if it is in an accept state when the input has been completely consumed, others require that the stack be empty when the input has been consumed; some require a single accept state, some don't. All these models are equivalent, but you need to look closely at the definition you are using.