Yes this has been very informative.  New questions:

There seems to be at least 3 cryptographic elements

1- addresses have public and private keys, they can be easily generated but doesn't seem like there is any reason you couldn't have duplicate addresses, other than highly improbable.

2- the nonce within a block.  It seems like this has to be found like mining.  Do most miners mine for nonces?  Once someone discovers the nonce for the current block the block is finalized and search for a new nonce begins.

3- bitcoins themselves.  These seem to be some other key that miners search for and seems to be independent of the nonce.  Once a coin is discovered (mined) it is put in a block so the miner has ownership, it is assigned to the miners address.

Does each new block contain one and only one new bitcoin along with a bunch of other transactions?