Will there be any way to limit the amount of storage required as time goes on? Will blocks always have to go all the way back to 2013, or could they in future be based on a more recent reference point?
You can enable pruning. Pruning means that Core will delete most of the blockchain once it has verified it. It does this on the fly so you won't have the whole thing stored on your disk. It does still require you to download the entire thing, but old blocks are deleted after they are verified and a certain number of newer blocks are found on top of it. With the default pruning setting, the disk space required is reduced to ~5 GB. Note that you will still be consuming the same amount of bandwidth and still wait as long for the blockchain to sync when pruning is enabled. It does not speed that up.
To enable pruning, find the bitcoin.conf or make one in the Bitcoin Core data directory. Then add the following line to it:
When you start Core again, it will prune the blockchain.
I've never sold any Bitcoin. Presumably Core won't allow me to sell any until I've completely caught up on the 13 weeks I'm still behind. Is that correct?
Bitcoin Core does not have any functionality to sell Bitcoin. It allows you to send Bitcoin to other people. In order to send, you have to be fully synced.