...and during that process I wondered if anyone running a full node can confirm any Bitcoin transaction in a block even if they do not receive the gas fee? Is it necessary for the transaction to be confirmed by a miner?
Only miners can mine Bitcoin blocks, but mind you that miners do not mine a single transaction, the transactions will be compiled into a candidate block and then the miner will then try to add that block (containing many transactions) into the blockchain, the more transactions in a block will earn the miners more in transaction fees, but nonetheless they cannot exceed the block size limit.