Is energy affected by forces? Curious, is any certain form of energy, such as heat, affected by a force? Can you push energy from one place to another?
Update: An example case: Now suppose we have a solid medium by which energy can propagate. If there is energy in a specific region of the solid, can I move that energy to another specific region of that solid? How?
 A: If you heat that rod, its mass will grow, and thus its weight too. But the difference is unmeasurable low (practically, the rod will lost much more mass by its fastened vaporization). You had to heat it to milliards of Kelvins to grow its mass with some thousadths.
Such things are handled by the EFE (Einstein Field Equations). These equations are defining the relation between the curvature of the spacetime and the mass in it. The curvature of the spacetime is what we sense as gravity in our circumstance. And every form of energy has its mass, based on the mass-energy equivalence principle, this is the well-known $E=mc^2$ relation, which essentially says: $1kg = 9\times 10^{16}J$.
This black hole thing on the end I simply don't understand, I suggest to fragment that into smaller questions and ask them more elaborated, again.
Extension after question edit:
The EFE describes the relation between the energy density and the spacetime curvature. Here counts any type of energy, thermical energy just as normal matter (which is counted by the mass-energy equivalence). Thus the answer is yes, gravitation affects energy. Although if you calculate, how many energy has the mass of a rod weighting $1 kg$ ($9\times 10^{16} J$) or, how many thermical energy will it have if you heat this with $1000K$ (some ten-thousands of $J$), you can see that in normal circumstances only the rest mass which really counts.
In case of the other interactions, the situation is more complex. It is handled by the quantum mechanics, which doesn't operates with forces, but with virtual bosons. You can imagine that as if, for example, the attraction and the distraction between a proton and electron were handled by photons emitted by the particles.
