Does that vector also have fixed magnitude?

It's magnitude is called 'mass' (or 'rest mass'/'invariant mass' if you want to differentiate it from the concept of 'relativistic mass', which probably should be retired).

This means we have

    four-momentum = mass * four-velocity

whereas

    three-momentum = gamma * mass * three-velocity

Not sure if you will find this comment, but I'd be very curious to hear what the magnitudes of "four-force" and "four-potential" are as well!

My layman's research suggest these are, respectively, "electric charge" and "current"?

That doesn't sound right to me. Note that four-force is a generic concept not limited to electromagnetism. Current is the source term of Maxwell's equations, which connect it to the potential via the d'Alembert operator.

There is indeed also a relation between momentum and (rest) energy describing the conservation of energy, In the rest frame this relation reduces to the famous E=mc^2.

So Feynman diagrams describe rotations?