Yes and no. Temperature is a thermodynamic, bulk property, so it's at least disingenuous to speak of temperature of a lone particle. The connection between temperature and particle energy comes from statistical mechanics, which finds E = k_B*T, so it's not completely unfair!
Absolute zero doesn't mean "take away zero point energy", which is impossible of course -- the energy at zero is the energy at zero, that's how it's defined.
Motion doesn't need to stop, just the exchange of it with surroundings. The lowest energy state of a system can still have multiple ("degenerate") states with equal energy levels. Particles can exchange energy with each other, as long as the net change is zero.
Example: liquid helium
doesn't actually freeze, it remains a superfluid (though it does freeze under pressure). A superfluid does not dissipate energy as viscous shear or turbulence, it conserves it (apparently as quantum vortices).
Superconductors retain current flow, at zero voltage drop mind, and therefore retain a magnetic field too. Which is fine, because a magnetic field isn't an exchange of energy either, magnetic field is conservative. Or capacitors which remain charged, assuming zero leakage. Electric field is conservative, too.
As for the examples, energy corresponds to temperature, so an alpha particle at a few MeV isn't going very fast (due to its large mass), as particles go, but it is at quite a high equivalent temperature! Electron velocity in conductors is much higher due to thermal energy (~10^5 m/s) than to drift velocity (~cm/s), even at high current densities.
Velocity saturation can occur in some materials, where the electron density is small -- semiconductors -- and the breakdown E-field (impact ionization threshold) is high. GaAs is such a material. Entering this regime causes negative resistance (current decreases as voltage rises). This acts essentially as fast as the electric field itself getting into the material -- an easy way to generate microwaves (10s of GHz) with a small hunk of homogeneous material! Although, despite being homogeneous, they are called "diodes", when they'd really be monodes... of a sort. Gunn diodes to be exact.
Again, amps or volts aren't a measure of energy, their product with time is (E = V*I*t). You can have voltage without current, or vice versa, and consume no energy (or, slosh energy back and forth in a resonant circuit*).
*Even superconducting resonators have nonzero losses, actually. Pretty good, with a Q factor up to 10^8 or thereabouts -- better than quartz crystals -- but still a time constant on the order of seconds, definitely not everlasting.
Tim