In short, how would you summarize what is required for and would constitute a (transaction) confirmation
To "confirm" a transaction (to give it it's
first confirmation), a miner verifies that the transaction conforms to the rules of the protocol, then they package the transaction together with other verified transactions into a block generating a block header that is unique to that list of transactions, then they search for a nonce that results in the hash of the header being lower than the current difficulty, then they broadcast the block for peers to validate and add to their blockchain. When a transaction is first included in a block in the blockchain, that transaction has its first "confirmation".
and does it require a certain number of miners to "confirm" it?
Only 1 miner will confirm any particular transaction.
I understand that the number of confirmation of a transaction is merely the number of new blocks that have been generated after the transaction is "posted"(?) to the blockchain, so is it because the more new blocks posted since a given transaction that it quickly becomes extremely difficult for that given transaction to be altered in the blockchain and hence the transaction is nailed down securely for good into the blockchain?
Correct. The hash of the transaction is included in the block header as a merkle root (in other words, change any transaction in a block, and you change the header of the block). In doing so, the nonce that was published with the block would no longer result in the published hash of the block header. To convince peers to replace the block in the blockchain, an attacker would have to mine a new block that has a low enough hash value. Each block has in its header, the hash of the previous block, so to change a block that has a new block added after it, an attacker would have to re-mine both blocks to find valid new nonces for each. The further back an attacker tries to modify a block, the more newer blocks the attacker has to re-mine to convince peers to accept it. Meanwhile, the rest of the honest network is mining new blocks, so the attacker has to re-mine the old blocks faster than the rest of the network is creating new ones or they'll never catch up (and peers will only replace their current blockchain if the replacement is "longer" than the current one). Of course if an attacker could provide more hashing power than the entire combined honest network, then the attacker would be guaranteed that they would catch up eventally. That's typically referred to as a "51% attack".
Also, what does it mean when a block is rejected and how does that come about?
No sure what exactly you are referring to. If an attacker were to try to broadcast an invalid block, then all peers would simply refuse to relay it or add it to their own copy of the blockchain. If two miners find solutions for two different blocks (that each have the same block hash listed as the "previous block"), then peers will accept the first of those two blocks that are relayed to them. When they receive the second block that has the same "parent", they will reject it because it creates a chain that is the same length, and acceptance of replacements to current blocks requires a "longer" chain. This can result in a temporarily "split" network, where some peers have one block added to their blockchain, and others have the other block. Eventually a miner working on one side of the split or the other will find a "next" block. When this new block is broadcast, it will result in a chain that is now the "longer" of the two. Therefore, peers that had been on the other side of the "split" will reject (orphan) the block they currently have and replace it with the two new blocks of the now longer chain.
Through what mechanism in the BTC network does a higher transaction fee added to a transaction would prompt faster processing / confirmation to a transaction?
There is nothing in the protocol that requires a miner to include anyone's transactions in their block. A miner includes transactions for two reasons.
The first reason is that if no transactions ever get included then bitcoin becomes useless and the mined bitcoins are therefore useless. The miners add value to all bitcoins by making them useful. The miners therefore add value to their own bitcoins by making all bitcoins useful. Now that miners have an incentive to add some transactions to the blocks that they create, the question is which ones to add?
The second reason is that they get paid to do so with transaction fees. Transaction fees provide an incentive for miners to add a particular transaction. In the beginning there weren't many transaction fees. Miners had very few transactions to choose from, and plenty of space available for those transactions within the one megabyte limit that the protocol imposes on each block. As such, miners were happy to include all available transactions in the block they were working on just to make bitcoin useful. As bitcoin becomes more popular, there can be more unconfirmed transactions than available space within the one megabyte limit. At that point, a miner has to decide which transactions to include in the block they are working on. By adding a transaction fee, you provide a financial incentive to the miners to include your transaction in their block instead of some other transaction that paid a smaller fee. All the transactions that don't make it into the next solved block have to wait unconfirmed until some other block where there is space for them. Unconfirmed transactions are easier for an attacker to modify since they aren't protected by the hashing work that miners do, so there is an incentive to users to get their transactions confirmed as quickly as possible.
