Does a PDA immediately accept if at final state with empty stack?
There are two ways in which a PDA may accept:


Final state: The PDA has finished reading the input and it is in the final state
Empty stack: The PDA has finished reading the input and its stack is empty.


The PDA in the example you have provided is non-deterministic, and when non-deterministic systems read an input for which a transition is not defined, they get stuck. In this case, when the PDA reads the string aaabbba, it would get stuck after reading the last b.