miners, afaik
Correct - along with the (currently) 25 BTC block reward the successful miner also gets the fees for all txs included in the block.
For renting your resources (CPU, GPU, Electricity) to the network? If I had a powerful computer, I would have been a miner too.
By the way, here's why I thought fees are not always needed:
Transaction fees may be included with any transfer of bitcoins from one address to another. At the moment, many transactions are typically processed in a way where no fee is expected at all, but for transactions which draw coins from many bitcoin addresses and therefore have a large data size, a small transaction fee is usually expected.
(From en.bitcoin.it/wiki/Transaction_fees)