If you heat atoms/molecules well enough, can they exceed the speed of light? Or is there also an opposite of absolute zero at which all movement stops?

Answer

You can accelerate charged particles until they approach the speed of light. That’s exactly what happens in a particle accelerator. This only works with charged particles such as protons, electrons. Accelerating naturally takes a huge distance and so you have to make the particles move in a circle. The carefully tuned and precisely changing magnetic field of an accelerator thus has a double task: keeping the particles in the circular orbit and accelerating the particles.
As mentioned, speeds that are only a fraction below the speed of light can be achieved with this, but the speed of light itself cannot be achieved by particles with a mass other than zero. It can therefore certainly not be exceeded. To accelerate a particle with mass to the speed of light itself would require an infinite amount of energy.