AES block cipher modes of operation
AES has a 128-bit block size, period. The valid key sizes are 128, 192, and 256-bits.
I am not sure what you mean by that, a "block" is a 128-bit group of data, being the input and output to the cipher, it does not receive anything. Please revise that part of the question, and I will adjust my answer.
CTR, OCB, and ECB are the only modes I know that can run encryption on blocks independent of eachother. CTR is the obvious choice for a fast secure implementation and turns the block cipher into a stream cipher. OCB is a more complex mode built on ECB, where both the input block and cipher output are modified using an incrementing counter generated from a nonce, and provides 1-pass authentication.