Are these two events mutually exclusive? 
*

*Let's consider event A and B where $\Pr(A)\neq 0$,$B=\emptyset$.

Are these events mutually exclusive since $\Pr(A\cap B)=\Pr(A\cap \emptyset)=0?$


*What if event A = event B and $\Pr(A)=\Pr(B)=0$ but $A,B \neq \emptyset$

Are these events also mutually exclusive since $\Pr(A\cap B)=0?$
But for the 2nd part, $A\cap B =A=B\neq \emptyset$, so not mutually exclusive?
 A: If two events are mutually exclusive then it is impossible for both to happen. This is stronger than saying there is probability $0$ that both happen. So the correct condition is $A\cap B=\emptyset$, not $\Pr(A\cap B)=0$. In your first case they are mutually exclusive and in the second they are not.
A: Mutual exclusivity is defined to mean that the intersection of the events is empty.
A consequence of the definition is that if two events are mutually exclusive then the probability of their intersection is zero.  The reverse however is not true.  It is possible for two events to have the probability of their intersection be zero while not being mutually exclusive.
"Are these events mutually exclusive since $\Pr(A\cap B)=0$?"  Not for that reason, no.  The first will be mutually exclusive since every event is mutually exclusive to the empty event.  The second will not be mutually exclusive since we are told $A\neq \emptyset$ and that $A=B$, hence $A\cap B\neq \emptyset$ as you correctly state in your post.
