Existence of gravitons? For a lot of my uninformed life, I have doubted the existence of gravitons or even that gravity is an actual "force" (like electromagnetism). This is because my vision of general relativity was that mass curves space such that objects are still traveling in a "straight line" when being acted upon by "gravity", so that no "force" is needed. I know now that this is a naive view, but I'm not 100% sure why. I was thinking the other day that just the fact that gravity follows an inverse square law implies that it is a force carried by particles (falling off in flux intensity due to the geometry of 3D space). 

My question would be: Does the fact that gravity follows an inverse square law naturally fall out of general relativity equations or is it an assumption used when developing the equations?

And, just now, I had the thought that other forces may curve space also (just in higher dimensions).
Gravity is a fictitious force, actually, much like centrifugal force.  In a free falling frame of reference it disappears.  In general relativity (GR) gravity is just a result of (differential) geometry: space-time curvature.  The inverse square law is just the low energy approximation, but the actual equation for gravity derived from GR is more complex than that.  The massive success of Newtonian gravity tells us that any model of gravity must be approximated by the classical inverse square law at low energies.

Whether GR does that by (Einstein's) design or something else is a matter of personal opinion.  Einstein definitely knew he had to get approximately Newtonian gravity at low energies, so he would have discarded or modified any ideas that failed this criterion.  However, there are standard arguments for why gravity must obey an inverse square law, at least in low energy situations.

Now in GR a lot more than just mass contributes to the gravitational field.  Spin and electric charge, for example, and quite importantly: energy (the famous $E=mc^2$ tells us how to express a mass as energy, so we can treat these things on a common footing).  So, yes, all forces and particles contribute to gravity.  Even photons.

GR itself makes no predictions (or requirements) for the existence of any new particles outside of the standard model, such as gravitons.  GR and quantum mechanics (QM) are famously incompatible: in extreme situations where both GR and QM are relevant (neutron stars and black hole formation, for example), they stop making sense rather quickly.  Especially GR.  "Gravitons" and assorted variations are hypothetical particles that are proposed to resolve this issue by creating a quantum theory of gravity.  The only "evidence" we have for them at this stage is that our two most massively successful theories about the workings of the universe, GR and QM, are so painfully incompatible.  So we know these theories are flawed (aka wrong) and that some other theory is necessary that can handle these situations, while also incorporating all of the successes of QM and GR&mdash;they're amazingly accurate when only one of them is particularly relevant, after all.

Exactly what that theory is is an ongoing and substantial research area.