A zero-fee transaction will still get confirmed, although it may take a day or so. Although this is certainly longer than it used be as mining gets more commercialized, this really shouldn't be the case since the 25BTC reward overwhelms current transaction fees so miners shouldn't care. It's anyway a good habit to give a fee because this is the only way Bitcoin can work long term.
Yes this is true. As the reward continues getting halved, the built in rewards of the system will go down. These fees will be where the source of mining income comes from when all of the bitcoins have been mined.