To get to the inside working of $E=mc^2$, we first need to understand what it actually means. Einstein wrote it in his paper like "$m=E/c^2$",why?
Because according to me when you write it as $E=mc^2$, you sort of think, 'hmm I think this equation says that mass is a kind of energy or that matter can be converted into energy", and that is really really misleading, in fact what this equation says is that mass is an intrinsic property of all kinds of energy, the inertia content of a body depends on the energy content of the body, the more energy you have, harder it is to push or pull. So if you have a ball having a mass $10 kg$( I will explain this $mass$ later ) and you kick it so that it gains some kinetic energy , now Einstein says that the moving ball has a high inertia as compared to the stationary ball, and why is that? Because it simply has more energy content, now all that kinetic energy is not being converted into matter, but it is $mimicking$ the original behaviour of matter, and it appears to us that mass has increased. Now you may ask that where does that $10 kg$ mass come from, is it $fundamental?$ The answer is still- "NO" , what's fundamental is energy, and that $10 kg$ mass is just the outcome of "atoms interacting with $Higgs field$".
Now your questions :-
1. What energy does $E=mc^2$
It gives energy much like all the other Energies, for example in nuclear fusion, protons and neutrons come together and form a nucleus, right ? And there is a huge energy release, now some people say that this happens because mass is being converted to energy, but that's not correct, see when you bring a proton and neutrons together, their potential energy decreases due to the "strong nuclear force attraction", see energy content is getting low, so due to this it appears to us that mass has been decrased, but that decrease occurred due to the lowering of energy content, and that lowering in kinetic and potential energy was released as heat and light, see it's energy that's being converted not mass. Mass itself is a measure of energy content, in other words you are not getting energy from MASS, but you are getting it from another kind of energy.
Question 2 :-
To release the energy contained in matter, you just need to distort its stable kinetic and potential Energies, for example when electrons jumps from higher state to lower state, its potential energy decreases and is converted to light energy, now you can also visualize it like this, when it jumps from higher shell to lower shell, its mass decreases ( due to lowering energy) and that loss of mass is equally compensated by energy release but this picture is certainly not correct, all the conversions are going from energy to energy, you dont need any mass to energy conversion.
When you annihilate matter and antimatter, all their potential Energies and kinetic Energies are converted into light and heat, and mass becomes 0 because photons don't interact with Higgs field, so no Higgs potential means 0 inertia content.
Hope that answers your questions.