Can free bitcoins from faucets be actually spent? Suppose that I have a wallet with 0.001 bitcoins, from dozens of micro transactions (6000 satoshi in average). That means if I want to spend them, it must be concatenated and the transaction fee would be higher than the actual value?
You could consider creating a new address with 0.01 BTC on it, then sending the concatenated amount at 0 trx fee to each of the addresses containing the smaller amounts.

So: 


0.01 -> first address 
First address (now including the 0.01) -> second address
second address (now including the 0.01+first address -> third address


... ad nauseum.