The perhaps both tells you that despite the either ... or, this is ordinary or ($\lor$), not exclusive or ($+$). Thus, $q\lor p$ is exactly right.
To answer the final question, the string $\neg +(q\lor p)$ is meaningless: $+$ must appear between two propositions. You can negate something like $q+p$, but the result is $\neg(q+p)$. To see what it means, start with $q+p$:
Either classes are held on campus, or it is raining, but not both.
We want the negation of this:
It is not the case that either classes are held on campus or it is raining (but not both).
That’s not easy to grasp, however. We can improve matters by going back to $q+p$ and finding a better paraphrase:
Exactly one of the following statements is true: $(1)$ classes are held on campus, or $(2)$ it is raining.
The negation of that is clearly that either both of the statements $(1)$ and $(2)$ are true, or neither is. We can express that as follows:
Either classes are held on campus and it is raining, or classes are not held on campus and it is not raining.