energy conservation - What is the mechanism of particle anti-particle annihilation

My question is loaded with assumptions so to minimize them, I would like to ask it with respect to an electron and anti-electron annihilating.

When I think of annihilation, I think of electron and positron turning into energy where the total energy represents the combined masses plus kinetic energy each particle had prior to the collision and annihilation (Assumption 1).

But, doesn't the field of each charged particle also have an energy content and does this energy content of the charged particle field show up in the final energy? I have never seen this though, only the sum of the mass energy. (Assumption 2 possibly).

Also, how is it that the masses annihilate if they are not ANTI to each other. Mass has no opposite like charge does (or other things). Does the annihilation of the charges also somehow annihilate the masses and if this is the case, that would require energy so this energy should show up somewhere. Or is there some other opposite "thing" that results in annihilation of mass?

Physicists used to think that electron mass was actually the inertia of moving charge in a field (or something like that). Is there some kind of connection between mass and charge such that charge annihilation is also mass annihilation?