DooMAD did a great job of answering most of your questions. I'll try to add additional details without simply repeating what he's said (as much as I can).
The person who solves the Proof-of-Work for the first transaction gets rewarded with the coin value of the block (currently 50 coins but decreases over time).
Currently 12.5 and some time next year will be halved again.
As DooMAD stated, the block subsidy is currently 12.5 BTC, and will be cut to half of that next year. It is cut in half every 210,000 blocks.
However, the maximum allowed value of the block reward is the block subsidy PLUS the sum of all the transaction fees from all the transactions that are included in the block. The user is technically allowed to assign a reward that is less than the maximum allowable reward, but that would be a foolish thing to do (after putting in all that work to earn the reward, why would you assign a lower reward?).
I guess a lot of my confusion revolves around what a transaction is. Perhaps I have missed something but it seems to me that any documentation I have read seems to gloss over what a transaction is.
Transactions are merely a set of one or more inputs and one or more outputs. The inputs refer to the previous transaction and the outputs are an instruction on where to send the BTC and in what quantity. In plain English, you're telling the network what you'd like to send and where.
Transaction outputs are assigned a value and are locked with some condition that must be met in order to spend them. The most popular condition is to require an EDCSA signature that is generated with a private key whose public key hashes to a given value. The hashed representation of that public key along with an indication of the requirements is then what is called an "address".
Each transaction input is a reference to a previously unspent transaction output along with some piece of data that satisfies the spending conditions placed on that output.
Transaction inputs provide value to the transaction (the sum of all the values of the referenced outputs).
Transaction outputs provide the conditions by which that value can be spent by future transactions.
The sum of the value supplied by the inputs must be greater than or equal to the sum of the value assigned to each of the outputs. If the sum of the inputs is GREATER than the sum of the outputs, then any unassigned value becomes a transaction fee that the miner can assign to his own reward transaction when creating the block that he attempts to solve.
If blocks form a chain and must be solved, and if the solution of the block depends on the hash of the block, doesn't adding a transaction to a block invalidate the solution?
Yes.
This is why it is impossible for anyone to modify the historical record that is the blockchain. In order to do so, they would invalidate the solved block that they are trying to modify. They would then need to re-solve that block. However, any solution to that block will modify its hash and will therefore invalidate the next block in the chain. In order to modify any historical block, they would need to re-solve EVERY BLOCK THAT HAS ALREADY BEEN SOLVED SINCE THAT POINT IN TIME. However, the entire time that they are trying to get caught up, the true chain continues to grow with more blocks. In order to ever catch up, the attacker would need to control more Bitcoin mining hash power than the entire rest of the world combined.
In the context of blocks and transactions, what exactly is a coin? If a newly created block has a single transaction containing 50 coins (as they currently do), how is a coin represented inside a transaction/block?
Coins don't exist in Bitcoin. Not in any real sense, and not in the sense that was proposed in the whitepaper.
The whitepaper is a good introduction to the larger concepts of proof-of-work, and how transactions can be represented, but when Satoshi actually wrote the code, he made some adjustments for a variety of reasons.
Now "coins" are just a word that we humans use to make it easier to talk about the transfer of control over value.
Also, a newly created block contains MANY transactions, not just the one special transaction that provides the reward to the miner.
The whitepaper says, in section 7: "Once the latest transaction in a coin is buried under enough blocks, the spent transactions before
it can be discarded to save disk space." The wording "transaction in a coin" confuses me here because I would have thought it should have said "block" instead of "coin".
As I said, the whitepaper is NOT a perfect description of how Bitcoin actually works. It is Satoshi's thoughts on how the concept could possibly work. The white paper was released months before the code. Not only did Satoshi make some modifications to the design after he released the whitepaper, but more changes have been made since the initial release of the code.