[Coda]

All fields of force have corresponding potential energy. This means it takes energy to move an object against the field lines, and that energy is released by the object moving along with the field lines. This is true regardless of what kind of field it is. The different potential energy formulas are descriptions of how that particular field works. (As far as I know, kinetic energy is just kinetic energy, and there aren't different forms of it.)

F = ma is not a field. It doesn't cause anything to happen. It's a description of a relationship, and it's true of ALL forces, masses, and accelerations. So it doesn't matter "what kind" of energy it is.

By convention, electric fields are described as moving from positive to negative. This is often considered to be a historical mistake -- Benjamin Franklin arbitrarily chose one of the charges he was able to create in his static electricity experiments to be positive, and the other to be negative, and all of the math from then on holds to that convention. It IS just a convention, and as such it doesn't actually MATTER because the whole point is that everyone knows the convention and therefore agrees on what things mean, but it means that the fields describe the motion of a positively-charged particle when in practice the particles in motion are negatively-charged. But to answer your question: Yes.

As for negative fields... No, those aren't possible. Vectors are a magnitude and a direction. While the math works out to consider a vector with negative magnitude, it's 100% congruent to a vector with a positive magnitude in the opposite direction. Negative lengths are considered undefined, because measures are positive by definition. And so, in the interests of keeping everything consistent and unambiguous and mutually understandable, it isn't done.