I'm fairly certain that learning more about mining and proof-of-work in general will help answer most of your questions, but I thought I'd get you started.

What does the process of finding unconfirmed transactions and submitting them into the blockchain look like?

This is mining. It means taking unconfirmed transactions, checking if they are valid, grouping them into a block, and performing a proof-of-work on that block. Miners don't have to "find" unconfirmed transactions; the transactions come to them. Anyone who is listening on the Bitcoin network will see new transactions as they are broadcast.

Is confirming transactions just as expensive as minting new coins?

Confirming transactions is mining. A confirmation is the inclusion of a transaction into a block. You may have meant validating transactions, meaning checking if a transaction is valid. In this case, no, validating a transaction is not an expensive operations. It's simply checking that the transaction conforms to a standard and is not attempting to spend money that has already been spent.

Does bitcoin core's bitcoind expose unconfirmed transactions?

Yes. You can use getrawmempool to see the transactions still in the mempool.

How do unconfirmed transactions get submitted into the blockchain?

The user who creates the transaction broadcasts it to the network, and then miners include it in a block.