So now I set faucetbox to send me BTC only if it reaches 0.2 BTC...
0.2 is probably much too high - you will virtually wait forever to get a payout.
0.01 might be reasonable, I think it would result in less than 1% fees for most cases.
The recommended fee is about 0.0001 BTC per kbyte of transaction data. One kbyte will normally be enough to handle a number of those small inputs, so for sending anything from 0.01 to maybe 0.15 BTC your fee would be 0.0001.
Onkel Paul
It is actually 0.00001 for nodes to relay your transaction.
If you want a free transaction, leave the inputs there for a while. If the priority is high enough and size is small enough, you can send a no fee transaction.