To better understand the below concrete answers, a short background on how disk encryption is usually performed with AES. The most tools will use one of the following two modes:
- CBC-ESSIV, which is just standard CBC encryption where each sector is one "message" and the IV is derived deterministically from the index of the sector to be encrypted as $\text{IV}=E_{\operatorname{Hash}(K)}(\text{SID})$ where $K$ is the AES key.
- XTS which is the more modern choice because it doesn't suffer from CBC's malleability essentially turns AES into a tweakable block cipher and using a different tweak for data block to hide equal blocks.
[How] does it relate to sector sizes on the drive which is typically 512 bytes?
The sector size usually must be a multiple of the block size of the underlying cipher so no space is wasted to padding and stream modes don't have to be used (which break even more horribly than CBC under chosen-ciphertext attacks).
[Who] or what is responsible for dividing disk sectors into data blocks?
The disk encryption software will usually divide the 512-byte blocks into 32x 16-byte blocks for the AES encryption.
Does it mean that I will use the same key to encrypt all sectors?
Yes.
Or maybe this key is used to calculate/derive sectors keys/block keys?
No. However in ESSIV mode (see above) a hash of the key is used to generate CBC IVs.
