As an experimentalist, other similar null searches are the search for proton decay, the search for fractional charge, the search for electric-dipole-moments (of electrons, neutrons, atoms, etc.), tests of lorentz symmetry, tests of the unitarity of the CKM matrix, the search for Majorana neutrinos, etc.
The lack of proton decay is first on that list for a reason. Nobody really knows why protons are so stable.
Well, that depends on what you take to be the defining novelty of the equivalence principle, no?
So there are other forces that scale like the mass, most notably fictitious forces -- like centrifugal and Coriolis forces. Indeed one way to describe the equivalence principle in general relativity would be to say "gravity is a fictitious force that you feel in coordinates that are not in the proper inertial state, which you'd call free-fall." Most of the other structure is just given by special relativity, which says that whenever you accelerate in any direction, clocks ahead of you seem to tick faster in proportion to your acceleration and their distance from you; and clocks behind, slower (until a sort of "wall of death" at x = -c²/a where clocks appear to stop ticking, a so-called "event horizon"). To first order this is the only thing that is novel about special relativity and all of the other effects can be derived from it. So if you are listening to the equivalence principle and it says that you are accelerating upward by virtue of remaining on the surface of a planet, rather than being in free-fall downward, then you can immediately understand that there must be a gravitational time dilation of your clocks relative to the clocks out in space that you see ticking faster, and in turn you would also see dilation of the clocks that are at a lower altitude than you are.
If you don't find that terribly novel, you might like some various other ideas that say "these two things that you think are different are the same." Electromagnetism itself is one; electricity seemed to be a property of wool and glass rods while magnets seemed to be hunks of metal that deflect compasses -- then Faraday starts moving compasses with coils of wire. Or for example, Maxwell famously added a correction term to one of the electromagnetic equations (they do not work too well if net charge accumulated at a point, for example in a capacitor, unless you invent a fictitious current to replace the real one). When he did he realized that there was a way to make waves in the electromagnetic medium, and that he could use the constants he had available to calculate what those would be -- and he discovered that they would travel at the same speed as the speed of light. So he inferred correctly that light is an electromagnetic wave, just at a much higher frequency. Then this happened again once more: Glashow, Salam, and Weinberg shared the 1979 Nobel for the discovery that the weak force can be nicely modeled if you suggest that at high energies it unifies with the electromagnetic force, but then at low energies it decouples from the rest of them (ultimately via the Higgs mechanism, which was developed at around the same time). So these two things that seem to be different -- radioactive decay and conventional chemistry -- turn out to fundamentally be two low-energy sides of the same coin, once we pick out the part of the electroweak field whose interactions with the Higgs field cancel to be "electromagnetism" and its orthogonal complement to be the "weak force".
Even that doesn't seem quite what you are asking. Here's another stab: we discovered that there is a very natural way to phrase the world in terms of "Lagrangians", and Emmy Noether famously proved that when you look at this world that way, there is a duality between continuous symmetries ("the laws of physics are the same from second to second") and conserved quantities ("energy is conserved").
No, that's a bit too abstract. I guess the question is, what exactly are you looking for?
