What can I do with Bitcoin Core to see my current balance? I've waited 4 days now for Bitcoin Core v0.12.0 to synchronise the network and its getting really frustrating, especially when I still have 2 years and 38 weeks worth of the blockchain to download. 

The thing is I previously bought some Bitcoins to the address provided by Bitcoin Core before I realised that the blockchain had to be fully synchronised before I could see the transaction in the wallet! I now am a little bit stuck; Is there a way to move that Bitcoin address from Bitcoin Core to a completely new wallet like Electrum? Or is there a way to speed up the process of Bitcoin Core synchronising?

Regards,

Jordan Ellis
Bitcoin Core incrementally shows transactions you received as it processes blocks, so if you received the transactions a while ago they should pop up soon. You will not be able to independently verify the validity of your transactions without fully synchronizing.

If you just want to check the balance of your addresses it might be sufficient to look up the addresses on a public block explorer. The JSON-API gives you various methods for retrieving your addresses from Bitcoin Core. Once you have the addresses simply check their balance using a block explorer.

If however you'd like to move to a different wallet, the backupwallet API should give you a copy of your wallet that other wallet software can import.