When I get btc from a faucet, do I pay the transaction fee? Or does the faucet pay it?
The faucet pays it. They group a bunch of small transactions together and pay one fee for the overy all transaction.
Where you get killed is that using all those small transactions in your wallet will cost you more than the Bitcoin Dust is worth.
Thanks. What if I take all that dust and send it all to a new address in a different wallet, will it all turn into one coin (or whatever amount is) since its a new transaction?
Yes it will, but since transaction fees are "per kilobyte", and each dust input into the transaction is nearly 200 bytes, you'll pay approximately 0.0001 BTC per 5 dust inputs that you are trying to combine. If each input is less than 0.00002 BTC, then the inputs themselves won't be enough to cover the transaction fee that it takes to spend them.
Adding more inputs to the transaction, just increases its size in bytes, which requires a larger fee, which eats up the value of the inputs that you just added.
If you are running a wallet that has coin control, you can take a single larger value input (for example 1 BTC), and combine that with a few small value dust inputs. As long as your large value input was received long enough ago, and you keep the transaction size below 1 kilobyte, you can send the transaction with no fee. Then wait for it to be confirmed, and repeat the process. Since you are sending the transaction without a fee, it will take a long time to confirm. If the large value input is about 1 BTC, then it will probably take a day or so to confirm. That means you could sweep together about 3 to 5 small dust inputs per day. If all the dust inputs were 0.00002 BTC, and you did this continuously without fail every single day, you could possibly sweep up 0.0365 BTC per year.