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Yes, not very often.
A miner can assemble a block full of transactions it made itself or from others that pay no gas. If the miner finds a nonce then the miner can commit that block.
However, consensus algorithms make it difficult for a lone miner to win the privilege of doing it. Mining pools are uniquely positioned to leverage transient miner advantages.
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but how exactly are all of these transaction fees sent to the miner who mines the block?
They are not really sent anywhere - they don't need to be.
When a block is mined by a given miner, the consensus protocol ensures that the rest of the network becomes (eventually) consistent, meaning that all other miners accept the approved block, and get on with ...
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