256 bit symmetric cryptography keys are a bit like picking one atom in the universe (10^80 atoms, or 100000000000000000000000000000000000000000000000000000000000000000000000000000000). Your opponent would have to test half of the atoms in the universe to have a reasonable chance of getting the right key.
Correct. Anything higher is an order of magnitude more computationally expensive to do for no real reasonable gain. Multiple layers of encryption get you there far enough. Better to dig deeper into other cryptography methods than try increase AES beyond 256. Its already rather insane how quickly decryption happens.
You can trivially modify the AES key schedule to have a key size of any length (ex. replace it with a hash function or a sponge construct) and have any number of increased rounds in the AES permutation. Performance impact will linearly scale with the number of rounds.
You can even have no key schedule at all and just make your AES key size in bits = 128 * num_of_rounds. This doesn't mean that the bruteforce complexity is going to be that high but that would hardly matter...
That's generally understood to be not feasible.