Why are so many forces explainable using inverse squares when space is three dimensional?

This is not paradoxical and it is not necessary for any physical phenomenon to a priori have to obey any particular law. Some phenomena do have to obey inverse-square laws (such as, particularly, the light intensity from a point source) but they are relatively limited (more on them below).

Even worse, gravity and electricity don't even follow this in general! For the latter, it is only point charges in the electrostatic regime that obey an inverse-square law. For more complicated systems you will have magnetic interactions as well as corrections that depend on the shape of the charge distributions. If the systems are (globally) neutral, there will still be electrostatic interactions which will fall off as the inverse cube or faster! The van der Waals forces between molecules, for instance, are electrostatic in origin but go down as $1/r^6$.

It is for systems with a conserved flux that the inverse-square law must hold, at least at large distances. If a point light source emits a fixed amount of energy per unit time, then this energy must go through every imaginary spherical surface we think up. Since their area goes up as $r^2$, the power per unit area (a.k.a. the irradiance) must go down as $1/r^2$. In a simplified picture, this is also true for the electrostatic force, where it is the flow of virtual photons that must be conserved.

The field line picture known from school might be helpful with that:

The surface area of the surrounding sphere (and not it's volume) determines the density of the lines sourced by a point charge, corresponding to the field strength.

These physical phenomena (gravity, Coulomb force) are forces caused by an object you can consider pointlike. That is, for the inverse square law to hold, the object emits the force uniformly in all directions from one point.

That means that at any distance (call it $R$) from the object, you'll feel the same force as you would anywhere over the surface of a sphere whose radius is that distance.

The surface of a sphere is $2$-dimensional, not $3$-dimensional, and its area goes like $R^2$. The larger the radius, the larger the surface of the sphere, and the further away you are from the source. So the strength of the source is inversely proportional to the surface area of the sphere.