Does a PDA follow a transition if the top stack symbol doesn't match?
TL;DR: No, it does not.

I would guess that your notation a, a -&gt; b means the syntax in a PDA transition such that if we read a character a from the input and we pop a character a from the stack, then we put a character b on top of the stack.

The notation of that transition that I'm used of is written as $$q\xrightarrow{\text{e,d;u}}r,$$

where the left side of the semicolon is what we read, i.e. character e from the input and character (or symbol) d from the stack. Right side of the semicolon is what we put onto the stack: u.

In order for your PDA to follow a transition, it has to match both of the conditions on the left side of the above notation. So, your PDA won't follow that particular transition if the stack symbol doesn't match.

What comes to your second question about having two different transitions with same input but different stack symbol, yes, that is possible based on nondeterminism, but your PDA still has to meet both conditions in the left side of the notation in order to move anywhere.

Hope this had any help!