Your asking good questions protonic_mass.
Your absolutely right that a proton at rest (and not accelerating) and at 0K (which is sort of saying the same thing) would not emit any EM waves. However, these protons would not remain at 0 Kelvin since EM waves coming from stars etc. would jiggle the protons around with their EM fields. Soon the protons would have a non zero temperature and would be emitting light.
Now about your question about moving 1 kg of water and protons. First, when you say move, I'm going to assume you mean accelerate, since you must accelerate a charge to get an EM wave. If you accelerate 1 kg of water, it won't emit any EM waves because it has not net charge (it has the same number of electrons and protons), so in the end your not accelerating anything with a net charge. If, however, you accelerate 1 kg of protons, then yes, you will cause the emission of EM waves.
Now answering the rest of your questions in the last two paragraphs of your last post:
It does not take both electric fields and magnetic fields to cause an EM wave. An EM wave is caused by an accelerating charge such as a proton or an electron.
Yes, protons have electric fields that point outward from the protons, and electrons have electric fields that point inward toward the electron. The electric field does not consist of electrons nor protons; it is caused by them.
Potential/voltage are confusing concepts that aren't needed to understand this situation. Suffice it to say that protons (instead of electrons) can be used to create voltage/potential differences. Again, though, I don't think these concepts help us in this situation.
I'm puzzled by your last question regarding only having magnetic and gravitational fields left. It looks like you may be thinking that electrons create electric fields and protons create magnetic fields (not a bad guess given the name, "electric" field). As I said above, both protons and electrons create electric fields (and both can create magnetic fields). If the protons or electrons are at rest with respect to you, you will only detect their electric fields. If the protons or electrons are moving at a constant velocity with respect to you, you detect a mixture of their electric and magnetic fields. Finally, if the protons or electrons are accelerating, you will detect their EM waves (you would also detect their mixture of electric and magnetic fields not in the form EM waves).