Are there any good "light" (read: doesn't require downloading the blockchain and the space and time associated with that) wallets for mobile or desktop (or both)?
Electrum is a great wallet IMHO.
That I can quickly and easily import a bitcoin core wallet into?
No, the way Bitcoin Core stores the wallet file makes it incompatible with every light wallet that I know of.
If not, how can I most easily get a file or even a piece of paper that's storing the data I need to be able to quickly spend or trade bitcoins?
If your Bitcoin Core client has a HD seed that is generated with it, 0.13.0 onwards, you can dump the xpriv key out and store it safely. You can use the xpriv in various wallets. The main downside being that you cannot change the password of the wallet. Anything below 0.13.0, you have to backup the wallet.dat every 100 transactions.
If you are worried about the synchronization time, I would recommend you to just use Electrum. The seed is easy and it offers a rather good security for day to day use. I've trusted it with all my Bitcoins.
If you're in an emergency, you can still spend the coins before it synchronizes fully. You have to create a raw unsigned transaction and sign it using the client.