Is the transaction time possible to predict based on the transaction fee? And is it possible that a transaction with a fee of 0 BTC gets handled right away if there is a low number of transactions in the network?
The transaction time is based on the integration of your transaction into a newly mined block.
Miners will prefer transaction with a higher fee (+some other factors like coin age, amount of Btcs,datasize), so they will include them first.
If there is a low number of transactions your transaction without a fee may be handled right away.
Btw: usual transaction fee of the qt-client is now 0.0001 (unless the Btc amount is made up by a lot of tiny transactions. If you collect dust from the free BTC sites and try to send all that the transaction will have a very big datasize)
You should be fine by using that fee.