Can someone explain double spending to me? I am restoring a backup of my Bitcoin Core wallet on a new PC. I downloaded the complete bootstrap.dat and placed it in ~/.bitcoin. I thought using the bootstrap would make things faster, but importing from disk seems just as slow. Oh well.

Anyway, currently Bitcoin Core has imported blocks through Jan 2014. I made bitcoin transactions after Jan 2014 that aren't reflected in my wallet. That is to say, I have a higher balance available in my wallet than I really should.

Could I double spend these coins by sending them to another bitcoin address that I own?
No you cannot. When you try to send a transaction to the chain which is not allowed by the chain (so sending more coins then you own) results in a invalid transaction. The case now is that your program didn't process your recent transactions, and therefor thinks that there are more bitcoin in your wallet then there really are. Otherwise it would be far to easy to doublespend: just reinstall Bitcoin Core and have a wallet which held a lot of BTC prior a specific date. If you are interested in reading more about doublespending (which I think you are not, just wanted to know if you were possible right now) you can read this.