Are there AES-128 keys such that first round key equals last round key? I am looking for AES-128 keys such that the last round key is exactly the first round key (which is simply the key). Any pointer to a proof that they exist or that they don't would be welcome.
In the same line, I am looking for last round keys which are the bitwise inverse of the key or some similarly simple relation with the key.
Not an answer but larger than a comment.
You can ask automated solvers to provide you with keys where the first and last rounds keys match.  Existing solvers likely won't terminate quickly on AES-128 for reasons mentioned in the question and comments.
AES-256, on the other hand, does yield results. For example the keys:

0x623cacba4d7fc6b7914b6bd000a2bcccf0967f268cbafde97ffc5bae4be5d2,
0xfea74943353b4d96d69457a2e6fbe56b1029ca48dc179d1967200df51cee7 and
0xf4e810b7b1c9253e75683304ac4f03f90be6534757600685706d8f6a97bee

all retain the property that expandedKey[0] == expandedKey[nrRounds].