Is it possible to setup mining pool that would do only confirmations?
The purpose of mining is to find valid blocks to add to the blockchain. A transaction is "confirmed" when it is part of a block in a blockchain that has some number of blocks after it. The number of blocks after it is the number of confirmations the transaction is considered to have.
Not quite. The new blocks are the confirmations.
> There are no "types" of mining, afaik.
Right.
When they do that, if they are the first miner to come up with the values required to confirm the transaction they are allowed to add a transaction of their own which "transfers" some bitcoins from "nowhere" to them, thus getting rewarded for their work. it is this that incentivizes people to "mine" but it's not the purpose.