CERN’s Large Hadron Collider achieves the hottest manmade temperatures ever

There is no relativistic limit to temperature. You probably calculated temperature by using the speed of the particle (and assumed it to max out a c), but at relativistic speeds this is wrong, you should use the energy of the particle instead.

The question is: Does particle A impart energy to particle B, or does it take away energy? Whichever is faster imparts energy to the slower one. It makes no difference that the speed is limited - one particle can always have more energy than the other, and can therefor impart energy to it. It's not possible to actually reach that speed, so no matter how fast a particle is moving, you can always increase its speed.

Time dilation should cause some very interesting artifacts though, I have no idea how to go about calculating what would happen. Length dilation is even more interesting - you may find your particles are so small that they pass each other and don't collide.

However it's not actually the particle that collides, it's the electric fields that collide. But electric fields travel at the speed of light - so the fields won't really have time to notice each other.

And finally at these energies a collision is unlikely to bounce, instead it would create some new particles, using up the energy and cooling rapidly.

Technical limitations aside, not really. However, there's the Planck temperature of 1.42×10^32 K where our physical models break down.

According to wikipedia (heh), quark-gluon plasma is not really a plasma so much as a liquid. Presumably, that means the modes are still roughly momentum modes, and you can literally think of a hotter QGP as faster quarks and gluons sliding around and slamming into one another. It's a Fermi liquid, though, which means the exclusion principle has significant effect. (I'm a physics grad student, but not a high-energy theorist, so take that for what it's worth.)

NIF is applying a large amount of energy to a small but macroscopic amount of matter. The LHC and RHIC are colliding mere nuclei, so the amount of matter is roughly 22 orders of magnitude less. The LHC and RHIC collisions are vastly higher temperature (energy per particle) than NIF, but the total energy is vastly less.

Presumably, the temperature is inferred by measuring the various energies of the outcoming particles (just like in a normal collision) and noting that they are distributed thermally (i.e. according to Bose-Einstein or Fermi-Dirac statistics). This distribution is tell-tale evidence that the particles exchanged energy through scattering many times.

I guess you could call this thermal imagining, although "thermal imaging camera" denotes something that looks at infrared radiation and measures its distribution (in order to infer the temperature of the matter that radiated it). In this case, the particles themselves (and their decay products) are exploding outwards and having their energies measured.

There are some problems with describing the energy of a particle accelerator as a "temperature", but it makes the article more accessible and understandable, which is surely a good thing.