A very partial answer to address a misconception, since this is a multi-part question. You wrote: 

> *This makes me wonder if increasing the weight of the object on top of the atoms constituting the table, will cause them to be 'squeezed' or
> compressed, thereby increasing their energy state? If so, where would
> the source of energy pressing the atoms together come from?*

Atoms and solids and gravitational fields are inconveniently complicated, so I'll start by replacing this complicated system with a very simple system. 

Consider Alice and Billy playing on a see-saw. When Alice's separation from the planet decreases (the gravitational field does work on Alice), Alice doesn't need a rocket to keep her from smashing painfully into the ground - she just needs somewhere to put the energy that she's getting from the gravitational field. Alice puts the energy into the separation between Billy and the planet: when Alice goes down, Billy goes up. 

Back to the weight on the table. If the table and the weight were both perfectly rigid (no amount of force can make either deform without breaking) then, if we started the weight resting on the table, the gravitational field wouldn't do any work because the weight wouldn't move, so the weight wouldn't need anywhere to put the energy - the table could be perfectly fragile in addition to being perfectly rigid, and nothing would happen. That, of course, is not how reality works: perfect rigidity is a frequently useful counterfactual, not a property of real objects made of atoms connected by inter- and intramolecular forces. 

So we have a deformable weight on a deformable table, and we apply the force of gravity, which starts deforming the weight and the table. This allows the center of mass of the weight to go down: the gravitational field has done work on the weight. 

Just like Alice and Billy, we don't need a rocket expending energy to push in the opposite direction to keep the weight from falling through the table. We just need *somewhere to put the energy*. For the weight and the table, the place where the energy can go is into the intramolecular force fields holding the solid together. Much like a spring (or, a bit closer to reality, like an incredibly complex system of countless, constantly jiggling, tiny interconnected springs), the solid has a higher energy state when it is deformed by an external force than when it is resting naturally. 

The energy to cause the deformation comes from the gravitational field, which is a mathematical representation of the energy associated with the configuration of centers-of-mass in the system. 

Stopping the deformation does not cost extra energy (unless you do it with a rocket). It demands a repository to put the energy into. The energy that comes from the configuration of centers-of-mass in the system goes into the configuration of interacting molecules in the system. If the configuration cannot change in such a way as to hold that much energy in that amount of time, either the table or the weight must break.