As long as bitcoin exists, mining won't "finish". Although we may decide to call it something else in the future, the process will continue the same.

Yes.

Mining is a name we use for adding transactions to a block and solving a hash of that block with an appropriate difficulty.  "Miners" get paid from 2 places.  They get all the transaction fees for all the transactions that they choose to include in the block, and they also get newly created coins called the "block subsidy".  The block subsidy is cut in half approximately every 4 years until it is less than 0.00000001 BTC.  The transaction fees are expected to continue to grow in value as bitcoin gains popularity. Eventually the transaction fees will regularly be more than the subsidy, and finally the subsidy will shrink to 0 and the reward for creating a block will consist entirely of transaction fees.  The process of creating the blocks (what we currently call "mining") will be the same, but the only reward will come from the transaction fees.