So, the chain contains a full history of transactions, and a coin's current owner is determined by all nodes as being the last signature in the chain associated with a coin. Am I reading this right?
Yes. Whoever has the private key that can create that last signature can spend the coin-- he or she (or them-- eventually maybe as bitcoin clients add features) are the owner.
If the chain has a full history of transactions, including the most recent owner, then why can coins not be recovered from the chain? Say my wallet gets wiped - the chain still knows the ownership of my coins is with me. I suppose my key would have gotten wiped as well, and then I couldn't verify that I am in fact....me?
For that matter, if the ownership of all coins can be derived from the chain, what is wallet.db besides a person's key? Why not just store the keys, and derive coin ownership while online?
This on-the-chain identification of ownership could provide a solution to the deflation problem - if its a problem at all. If a given coin's final client had not been seen in....five years, say...then the individual is obviously not using bitcoin for monetary purposes, and the coins associated with that key could be liquidated and put up for generation once more. Of course, that deflation is a problem at all is very suspect, especially given what I've recently been told about how precision can change. However, I'm just thinking though technical options here.
Additionally, could people use keys not generated by Bitcoin? For instance, I have a PGP key with which I sign crucial files, and this key is backed up to the n-teenth time on any number of media. If I could use bitcoin by simply inputting this key into the software, and then the software could derive which coins are mine from the content of the chain, it would make losing access to coins far more unlikely. Obviously, switching keys would be akin to switching wallets.