How do faucets/micropayment-based services avoid/minimize transaction fees?
You have two questions here, so I'll answer them separately.  As far as how do faucets they avoid transaction fees, the answer is that they don't.

Most of the faucet sites I've used use a service called Coinbox.me which collects requests for payments and bundles them up into one large chunk of payments.  This is how they minimize transaction fees - they don't send all the microtransactions at once, they wait until they have quite a few and then send them out.  I've looked at these transactions before when they're going through and they do have transaction fees that are being paid that is, as you said, larger than the amount being sent out.  But I have had transactions where it is being sent to at least a hundred addresses, so while one small amount may far less than the transaction fee, the total amount being sent out is significant enough that the service can pay a transaction fee while still paying out these microtransctions.