How did inflation add energy to the universe? What mechanism did this occur by? In other words, where did that energy come from? Was it due to the quantum fluctuation (or that scalar field rolling down a potential energy hill) to a lower, more stable energy state? This energy then manifested as exponential expansion, and then switched from expansion into heating (by filling up the universe with hot quark-gluon plasma)? Do we know the mechanism for this expansion->reheating switch?
The usual argument is that the net energy of the universe is zero because the positive energy of the matter and photons is balanced out by the negative gravitational potential energy. This would apply to the inflaton field as well. This has always felt vaguely unsatisfactory to me, but I don't know of a better explanation.
The "expansion->reheating switch" is dependant on the model of inflation used. In the original theory proposed by Guth inflation ended by quantum tunneling into the broken symmetry phase, so there was a nice clear physical process. Unfortunately this gave predictions that didn't match the observed universe, so it was abandoned. As I understand it it, current theories just tweak the potential to make inflation end without giving any nice physical reason for the exit.
You're obviously interested in the subject as you've asked a couple of questions on it. I would start with Guth's book as it's well written and an interesting read. The book is a bit out of date now, but it will give you a good starting point for further reading.
In General Relativity, energy momentum flows from one region of spacetime to another. But there isn't necessarily a natural "total energy of the universe." It might help to contrast General Relativity with other theories.
In Newtonian mechanics, a particle might gain kinetic energy while a corresponding gravitational potential energy decreases, thus you get that kind of conservation of energy. The total energy is the same before and after any event. However, the amount of energy depends on who's looking.
In Special Relativity a transfer of energy has to happen at an event (a specific time and specific location), so you have to have changes in kinetic energy be compensated by a loss of energy in another body or field. An example is the electromagnetic field which has an energy density, momentum density, and stress at every point in spacetime. Energy can transfer from the electromagnetic field to the particle and thus you get conservation of energy. It can be expressed by saying the energy in some region of space at some time is equal to the energy at an earlier time plus or minus the net flux of energy in or out of that region of space during that time interval. But energy conservation is just one part of a unified energy-momentum conservation, and that conservation can be expressed in a frame independent manner.
In General Relativity you generalize the kind of conservation of energy as is found in Special Relativity, but the tensor T, called the stress-energy tensor, that keeps track of the energy density, momentum density and stress at every event in spacetime is actually the same tensor from Special Relativity, and so it has no terms that correspond to gravitational potential energy. Breaking the stress-energy tensor into just an energy part is frame dependent and General Relativity is formulated in a frame independent manner. Some people try to make an energy psuedo-tensor, but that is a different tensor. And it is the stress-energy tensor T (not the psuedo-tensor) that is the source of the gravitational curvature, just as charges and currents are the source of electromagnetic waves.
So simply put, don't expect General Relativity to have something like "total energy of the universe", because that's just something that isn't naturally there. There is a stress-energy tensor, which if you pick a frame gives you an energy density at an event, but there is usually no natural frame, so no natural energy density.
But when talking about the stress-energy tensor in different epochs, there might be a sense where is has certain properties and at other times has other properties. One property a stress-energy tensor can have is whether it satisfies various so-called energy conditions. And a common consequence of many energy conditions is that the energy density (for every frame) is non-negative. So the question about whether a particular stress-energy tensor has a negative energy density is a legitimate question in General Relativity.
If you add a scalar field to drive inflation that has a negative energy density in some frame, then it can be a source of energy for other forms of stress-energy, so the inflation field's energy density can go towards a zero energy density while the energy density of other forms of stress energy increases, so there can be a flow of energy density even if the universe as a whole doesn't have a total energy.