How to track Starknet L1 costs part 2

Question

Posting a follow-up question for better visibility.

Here is the original thread: How to track Starknet L1 costs

I'm trying to track all the gas fees paid by Starknet on Ethereum.

Currently I'm looking at:

1. Memory page registrations
2. Merkle proof registrations
3. Fri-Merkle proof registrations
4. Verify and Register proofs

Similar what can be found on this Dune dashboard: https://dune.com/queries/2248341/3686770

Now I'm trying to understand if I should also track gas costs occurred from StarkNet Operator calls (0x2C169DFe5fBbA12957Bdd0Ba47d9CEDbFE260CA7) to the StarkNet Core Contract (0xc662c410C0ECf747543f5bA90660f6ABeBD9C8c4)?

Some question regarding the above:

1. Seems like calls from Starknet Operator to Starknet Core Contract are used to update L2 states to L1. Starknet carries the gas cost for these transactions. Is this correct?
2. Are there any StarkEx transactions involved?
3. Is there some other on-chain cost incurring for Starknet that is missing from above?

Looked into events & functions calls of the Starknet core contract. Tried to find info about this from the docs.