So, let's say we have a proton, and it is at rest, and we use the equation λ=h/p to find it has a wavelength of 0 because it has no momentum. Makes sense. Then we also use the equation E^2=m^2c^4+p^2c^2 to find its energy from its rest mass, then we plug that into E=hv to find that it still has frequency. But, shouldn't it not have frequency because it has no wavelength? If we use fλ=v_p, we get 0 for frequency. Someone help! I'm confused!

You're equating the Compton wavelength and deBroglie wavelength. They aren't the same thing.

With momentum p=0 the wavelength is not zero but infinite. An "immobile" particle would need to be perfectly delocalized.

 

This is an idealized case.Particles being more or less localized, they have a minimum of momentum and energy, as a consequence, just because they're waves. Hi Heisenberg.

 

The frequency of a particle is not absolute - which I find disturbing. It is exactly as relative as energy is. Depending on if you include the rest mass or not, the frequency change. But you may also neglect or not its potential energy within Earth's gravitation, its potential and kinetic energy within our Sun's gravitation field, our galaxy's field which isn't known, and so on. If the particle has a charge you may want to include an electrostatic energy, but electric potentials are relative, not absolute. And maybe there are forces we still ignore, so we neglect up to now some energy in this unknown field.

 

Only energy differences are observable and do matter - and so for particle frequencies.