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)? That I can quickly and easily import a bitcoin core wallet into?  

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?

I want to store bitcoins offline, but if something happens in the market I want to be able to move them, spend them, trade them, etc. quickly.  Having to install a fresh bitcoin core client and wait for it to sync is not an option.

Thanks.

What happens if I change my wallet password but somebody has a copy of my wallet that was made before I changed my password?  Can they still spend my bitcoins with the old password and the old wallet copy?