Is there a hash function for a collection (i.e., multi-set) of integers that has good theoretical guarantees?
If you think of sets as living in universe $[u]$, it is quite easy to solve your problem with $O(\lg u)$ update time. All you need is a fast hash function for a vector of $u$ numbers, with fast "local updates". 

Wikipedia/Universal hashing suggests $h(\vec{x}) = \big(\sum_{i=1}^{u} x_i a^i \big) \bmod{p}$, where $p$ is a large enough prime and $a$ is uniformly drawn from $[p]$. When you add or remove element $i$, you have to add/subtract $a^i$ from the hash code, which takes $O(\lg i)$ time using divide and conquer for the exponentiation. Since a polynomial of degree $u$ can only have $u$ roots, the probability of collision for two distinct sets is $O(u/p)$. This can be made very small by taking $p$ to be large enough (for instance, $p=u^2$ and you work in "double precision"). If the sets are much smaller than $[u]$, you can of course begin by hashing the universe down to a smaller universe.

Does anybody know a solution with $O(1/p)$ collision probability when hashing to range $[p]$? This ought to be possible.