I'm still downloading the Blockchain (which is going to take ages). My question is, should available Bitcoins be updated straight away launching QT (before the blockchain finishes downloading)?
No, your real available bitcoin will be shown only after the whole blockchain downloaded
in the mean time, what you are seeing is the
(historical) balance based on the latest block in your pc
when the wallet is fully synced, then it will show the actual available balance
Currently the total is 0.0000000BTC available. What's interesting is that I can already see all the transactions and comparing in/out it looks like I should have some BTC left over. Do I just have to let the entire blockchain download before available BTC is refreshed?
read the previous posts to decide on what you want to do next
but I would suggest, first you should check all your addresses' balances on public explorer