It is the conservation of mass-energy, not the conservation of mass alongside the conservation of energy. You have to think of it as a sort of slide scale where you can shift your proportions from mostly mass to mostly energy and back again while considering your masses as simply bundles of stored potential energy. When you're dealing with every day objects and ordinary interactions, the slide scale movements between mass and energy are negligible and you can ignore them. It's when you're dealing with stuff on cosmic or sub-atomic scales that it can make a lot of interesting things happen.

Matter has a baseline quantity of energy that you can completely convert it into. For example, combining a positron and an electron will totally eradicate both and release all of their rest mass as thermal energy. Mass-energy conversion is how nuclear power is possible.

Hawking radiation should be pretty easy to find info on. I think he won a Nobel for that.