Bitcoin-Qt wallet is taking forever to synchronise. Can I just use the same address elsewhere to collect coins? It will take several days before the wallet is in sync. I already purchased a bitcoin and am waiting for it to transfer into my account using the address that was generated by Bitcoin-Qt. Can I just re-divert the Bitcoin to a cloud-based wallet that doesn't need to synchronize?
Are you sure the syncing is not advancing? If you hover the icon in the lower right corner, you should see the block count. Can you see it increase?

You can export your client's wallet.dat file and import it into a new Bitcoin-Qt wallet or into another Bitcoin wallet, either using the wallet.dat file directly or by extracting the private keys from it.