You can already tell the bitcoin client to prune old blocks, because theoretically you don't need to keep any blocks more than a few hours old as long as you maintain a database with the last known ownership of every coin. In the future you could expect a setting to have the client ignore coins you don't control and crunch the entire db down to less than 1 GiB.

Yeah, but you still need to download all of them at least once to ensure there aren't missing transactions. The proposal with the UTXO set would allow you to only download headers+UTXO.

Yes, you do currently have to download the whole blockchain at least once. I can't remember if you have to save the whole thing before pruning begins or if it can prune as it goes along. Hopefully the headers+UTXO proposal will be implemented soon, as the total size of the headers is only 32 MiB (80 bytes per block) and the total size of the UTXO set is only 1.2 GiB. We've already dramatically reduced the bandwidth requirements with Xtreme Thinblocks, so storage cost is the major limiting factor for running a node.

Of every address. There is no such thing as a bitcoin.

Sorry, I have a bad habit of using "coin" to mean UTXO. You actually don't care about every address, only the ones with balance.