Find $k$ subsets containing a particular element quickly
Use a hash table, keyed on elements of $U$.  For each subset $S$ of $U$, say $S=\{x_1,\dots,x_\ell\}$, you'll add $S$ to the hash table $\ell$ times, once for key $x_1$, once for key $x_2$, and so on.  Now to find all subsets containing some element $e$, just look into the corresponding bucket in the hash table and you can iterate over sets containing $e$ very rapidly.

This is basically just applying the idea of "indexes" from databases to your problem.  It is straightforward and provides excellent time complexity, at the cost of modest space complexity (approximately doubling the space needed to store all of the subsets).