Gas is the accounting system used to work out how much ether the sender pays to perform transaction.
The ethereum yellow paper (appendix G) sets out the schedule of gas prices per EVM operation.
The transaction sender sets a value in ether that they are willing to pay per unit of gas.
The miners can survey the pool of valid transactions and pick the ones willing to pay the most ether such that the total gas is not over the block gas limit (presently 12500000).
They are free to pick any valid transaction (or even no transactions) within this.
As you might expect they tend to pick the ones with the highest gas price offer. See this visualization of the tx pool: https://txpool.zengo.com/#0,24h
They add any address they like to receive they block reward (before they start hashing). Presumably again they tend to pick an address they control.
In general, they amount of "computational work" done to perform the transaction computations is small compared to the work done in mining.
To your questions:
- The miner chooses which account gets the specified ether per unit gas used by the transaction.
- The miner chooses which account gets the block reward for adding a block, and any gas fees for adding transactions.