You cannot use Bitcoin-qt without syncing the entire block chain. Try using Electrum or Multibit instead. They work without downloading the entire block chain.

You're probably not fully verified with Coinbase if you're having to wait 4-5 days for a buy to go through. Go into your account and provide them with the info they want and you should be able to do instant buys. I think they just require a credit card and bank account to enable instant buys.