If $P(A \ \cup \ B) = P(A) + P(B)$, is it the case that $A$ and $B$ are disjoint?

No, all you can deduce is that $P(A \cap B) = 0$, because $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ It doesn't mean that $A \cap B$ is empty.

No for example suppose you have a uniform distribution on $[0,1]$. Let $A=[0,1/2]$ and $B=[1/2,1]$. Then $P(A\cap B)=0$ but $A\cap B=\{1/2\}\not=\emptyset$.