I'm a new comer to bitcoin but I am intrigued by the idea and the implementation. I've read the whitepaper twice now and scowered the wiki and the forums but I still find myself confused regarding some basic concepts. Hopefully I don't belabor your patience but perhaps this is a hint that maybe we need to work on the documentation a bit more.

My confusion revolves around the relationship between a block, a transaction and coins. Here are a statement of facts as I understand them:
Blocks form a chain.
Blocks form the backbone of the Proof-of-Work: a block must be "solved".
Blocks contain a header and a tree structure containing transactions.
The first transaction in a block is special. 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).
There is a finite number of blocks: 21,000,000
I assume there is only a single chain, starting at block B0 and eventually ending at block B21,000,000 - 1.

Question #1:
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. 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? If the number of blocks are finite and blocks are not mutable (transactions cannot be added to them), doesn't that imply that the number of transactions is also finite.

Question #2:
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?

Question #3:
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". Perhaps this would be answered by question #2.

I fully expect I am missing something here but I can't seem to find the answers in the existing documentation.