In the past it was possible to get a transaction without a fee confirmed if the inputs and outputs had a high enough priority and followed some rules. As of 2017 is a "free" transaction still possible if you don't care about how fast it will be confirmed? Several hours or even a day or two would be acceptable.
The Bitcoin consensus rules do not require transactions to have a fee, so it's still theoretically possible for a free transaction to be confirm.
It is, however, unlikely that any miners today will confirm free transactions. You can test this by sending yourself a free transaction to see if it confirms (but be sure you use a quality wallet, such as Bitcoin Core, that has features for handling transactions that never confirm).
Yes. It is possible. But what exactly do you want to achive? Either you want to minimize the fees or you want to delay transaction confirmation. Difference in details...