Are charged particles cold? Are charged particles colder than neutral ones?
If a charged particle is vibrating due to temperature, it will release some of its energy as electromagnetic waves. So that means it's losing energy, cooling itself off. Is there an error in my logic?
 A: Short answer: elementary particles, no. Composite particles, yes.
A charged particle just moving without acceleration does not lose energy, but if it accelerates then the Larmor formula applies (in the non-relativistic case): $$P=\frac{\mu_0q^2}{6\pi c}a^2.$$
So the relevant question is, will a hot particle accelerate?
If the particle is elementary even speaking of its temperature is problematic: usually what is meant is its speed or kinetic energy. So in this case it will just zip along without losing energy until it hits something.
Normally we speak of hot composite particles having a lot of mechanical vibrations going on, but these are internal. They do not produce a joint motion. So just being hot does not mechanically move the particle and hence cannot radiate away energy despite being charged.
However, the component particles are also charged and experience accelerations: they will radiate energy. At this point (for big enough particles, small ones deviate a bit) one can apply the blackbody and Planck's law reasoning to argue that thermal energy ends up stimulating electromagnetic modes, and energy will leave the particle as a whole. But now the overall charge of the composite particle is not doing anything.
A really highly charged composite particle may have electrons or ions expelled through vacuum discharge, losing energy that way. In fact, a hot charged metal may radiate away electrons as a thermionic discharge.
Charged particles have a few more ways of getting rid of heat than uncharged ones. But they will not generally become colder than their uncharged surroundings.
A: They are cold and lonely, and other particles of the same charge find them repulsive ;)
More seriously - cold/hot is the concept based on temperature, which is the measure of the average kinetic energy of the particles. As such, the temperature is not directly related to the charge of particles, although in some situations one may affect the other. One cannot also apply the notions of cold/hot to a single particle.
Also, accelerated particles emit light in classical theory (see Larmor formula), which is one of the reasons why this theory is problematic, and the quantum approach is needed.
A: There are two main errors:
Single particles don't have a temperature. Temperature is a statistical feature of bulk matter.
Single particles don't emit EM radiation when they move. Instead their energy is quantised. Under Classical theories all atoms would quickly collapse as their electrons radiate all their energy away.
A: Temperature is defined for a system in (at least local) thermal equilibrium. The electromagnetic field is present everywhere and, when in thermal equilibrium, has a blackbody spectrum. Thus, a body made of charged particles "vibrating" at temperature $T$, in the presence of an electromagnetic field at the same temperature $T$, will on average gain as much energy from the field as it loses to the field.
A body "will release some of its energy as electromagnetic waves" at a net energy loss only when the electromagnetic field is initially colder than the body. So yes, a body can cool (here by radiation), but only because it is not yet in equilibrium with its environment.
Meanwhile, uncharged particles simply lack a coupling to the electromagnetic field, so they can maintain a different temperature from that of the field. They do not necessarily stay warmer than charged particles. For example, if placed in a hot oven, uncharged particles would stay cooler because they are not heated by radiation.
