Are room temperature superconductors theoretically possible, and through what mechanism? At the moment, the highest critical temperature superconductor known to science (or myself, at least) is mercury barium calcium copper oxide. With a $T_{c}$ of roughly 133 K, that's well above the boiling point of nitrogen, and even well above the boiling point of oxygen, though using liquid oxygen to cool down anything probably wouldn't be the brightest idea. However, it's nowhere near the type of temperature that can cheaply be maintained, and far further still from the temperatures found naturally.
Are room-temperature superconductors forbidden by any known theory? If not, is there any known theory stating a mechanism by which they could operate, and what is the mechanism?
 A: Room-temperature superconductors are not forbidden by any known theory. However, discovery is difficult, while engineering is possible. One thing about superconductors is that they do not give off any heat. So cooling is just a function of fighting the insulation. With the discovery of super-insulators, the rest is just engineering!
Here's an interesting wikipedia article on the subject.
It should, however, be noted that these kinds of quantum states may be more common than non-quantum states. For instance, it is believed that neutron stars may be in a quantum state of some sort (perhaps super fluidity - I'm going from memory, so the details are a little hazy).
One thing that is clear is that with the application of extraordinary pressures, the transition temperature generally goes up. The highest pressures are supplied by diamond anvils, where the pressure chamber is formed between the points of two diamonds. Researchers generalise to other control parameters (Temperature, Pressure, Magnetic Field, ...), but generally speaking the control parameter is inimical to superconductivity (with pressure being the notable exception).
A: As mentioned here, metallic hydrogen may be a conventional superconductor up to about 290 K. This is then due to the low mass of the metal ions, this leads to a strong coupling of the electrons with the lattice vibrations.
