While reading about the Hamming (7,4) code, I saw that it was self dual. After looking up the definition, the dual code has a generator matrix equal to $H^T$, where $H$ is the parity check matrix of the original code.
But these two matrices are of different dimension! How can they be equal to each other?
Also, using the definitions for $G$ and $H$ from the wiki article, they are quite different (after transposing). I'm guessing that there is a large amount of non-uniqueness in these matrices, perhaps even when it comes to their dimensions. What would be the cleanest way to check that it is self-dual?
It is not true that this Hamming code, or any Hamming code for that matter, is self-dual.
A self-dual code must have even length, and its dimension is half its lengths. (This is precisely so that the problem with the dimensions of the matrices you point out does not occur.)
Yet, the length of the code is $7$.