Is thermal noise "quantum random"? Is the randomness that can be extracted from thermal noise "as random" (that is, even theoretically inaccessible to measurement according to our knowledge of quantum mechanics, and not just random for practical purposes) as the one that comes from "true" quantum phenomena, like radioactive decay of nuclei?
Due to the uncertainty principle, we can't predict both a particles position and speed infinitely accurate, and therefore we shouldn't be able to predict thermal noise, so it would seem that there is at least some level of quantum mechanics involved, but is that the same randomness that dictates the decay of nuclei, tunneling and other quantum phenomena?
So, is there a true distinction between thermal and "quantic" noise on a theoretical level, or are they basically the same? In other words, is the thermal movement of atoms and molecules fundamentally deterministic or indeterministic?
 A: In general, thermal noise is simply the noise due to a macroscopic point of view, which neglects microscopic (classical and/or quantum) features and only describes these statistically.
Thermal noise is not intrinsically quantum as it also arises in the statistical mechanics of classical systems. However, classical and quantum statistical mechanics often differ in their macroscopic predictions, which means that part of the thermal noise is due to quantum phenomena (i.e., cannot be explained on a classical basis). As everywhere when classical and quantum contributions must be distinguished, the classical part is that proedicted by the classical theory, whereas the quantum correction (the ''truly quantum part'') is the difference between the classical and the quantum result. Thermal noise is not special in this respect.
One of the typical forms of quantum noise is shot noise. See http://en.wikipedia.org/wiki/Shot_noise


The introduction ''In other words'' of your last question ''In other words, is the thermal movement of atoms and molecules fundamentally deterministic or indeterministic?'' is misleading, as you are not rephrasing what you had asked before. The answer to the last question is ''indeterministic'', according to the main stream interpretations of quantum mechanics. But this has nothing to do with thermal noise.
A: A common way of answering is something is a quantum effect is to see what would happen if Planck's constant was zero. In this example, it was Planck himself who did this in 1900 when he explained the observed formula for black-body radiation. Without quantization of the radiation, the formula would have an ultraviolet catastrophe. So without the exchange of photons for thermal radiation, the observed behavior would differ markedly for our current experience, as described by the attempted classical explanations for black-body radiation.
