Pseudorandom permutations on domains of arbitrary (i.e. not a power of 2) size
Yes. This is called format-preserving encryption.

The most flexible algorithm is FFX, which uses a Feistel network with AES-based round-functions, but performs addition modulo $m$. For certain values of $m$, the range of the round function is extended in order to limit statistical biases to negligible values.

When $m$ is very small, this approach isn't that great because its provable security gaurantees start to vanish if the adversary can see around $\sqrt{m}$ queries. If this is a concern, you can look at slower but more secure algorithms like Sometimes-Recurse Shuffling.