In $E=mc^2,$ does it not matter what constitutes the mass? I understand $E=mc^2$ calculates the amount of energy inherent in a given mass. Mass meaning "an object's resistance to acceleration (a change in its state of motion) when a net force is applied" (1) and Energy meaning "the property that must be transferred to an object in order to perform work on, or to heat, the object." (2) This Mass-energy equivalence makes more intuitive sense to me when thinking of those explicit definitions and how they are inherently related to one another and tied together through fundamental forces.
As I consider my question, I may just be unclear on the nature of mass and how it is independent of other properties. But I can't help to wonder - does a gram of sugar, a gram of soil, a gram of water, and a gram of lead all contain the same inherent amount of energy, because they are all of equal mass (or, said in a different way, because they are all of equal resistance to acceleration)? It could be that the only difference between equal masses of those materials are in other properties than their inherent Energy (for example, a gram of water and a gram of lead have different volume and conductivity).
 A: Very simply, no, it does not matter. But there is a bit of a twist.
At the most basic level, relativity doesn't care about the differences in the matter. I think it's more correct to say that it doesn't even enter into the equation, the "mass" in that formula is the inertial mass and that is independent of content.
But here's the twist... 
In that second part of the question, you're asking whether the energy inherent in one is different than another. And the answer is "yes", there are real differences. That's because it takes a certain amount of energy to hold a nucleus together against the mutual repulsion of their protons. So if you added up the masses of all the parts of, say, a carbon atom, you'll notice that an actual carbon atom weights less than that number. And you'd also find that an oxygen atom is even less.
But now the punch line: our measurement of mass is, because of that same equivalency, actually a measure of all the energy in that "thing". So while all of these objects you mention have really different internal states, at our human-sized scale that's what we call mass. So purely for that reason, a gram is a gram is a gram, and the E for that gram is a gram is a gram.
A: To answer your actual question:

Does a gram of sugar, a gram of soil, a gram of water, and a gram of lead all contain the same inherent amount of energy, because they are all of equal mass?

Yes, you are essentially correct. There are several types of what you call "inherent energy". In the ordinary matter, about 99% of it comes from the energy of gluons, the force particles (or more precisely the strong field) holding protons and neutrons together in the nucleus of atoms. The bulk of the remaining 1% comes from quarks (of which the protons and neutrons consist) interacting with the Higgs field. Then electrons (whose mass comes from other sources) add roughly about 0.05%. The total number of different types of energy contributing to mass is quite large (perhaps a few dozen depending on the situation) and may include the heat, chemical energy of bonds, and many other types, but again, the bulk of it is what is mentioned above. So the exact composition of different types of energy in lead and sugar may be slightly different, but overall only in a tiny fraction of a percent with the rest being essentially the same.
Please note that I am describing the normal everyday matter moving at ordinary speeds. There are things in the universe with mass from very different sources, such as black holes, neutron stars, relativistic objects (if you count the kinetic energy), etc.
A: They will have the same relativistic rest energy, since it depends purely on the mass. There can be other sources of energy present in an object such as thermal energy, but in a relativistic setting these are considered to be contributions to the mass of the object. Wikipedia explains this succinctly:

Chemical, nuclear, and other energy transformations may cause a system to lose some of its energy content (and thus some corresponding mass), releasing it as the radiant energy of light or as thermal energy for example.

