Now I know that temperature is the average kinetic energy of atoms, molecules and compounds. Photons increase the kinetic energy/temperature of molecules since photons are pure energy. Hence, wouldn't the transfer of kinetic energy mean that photons need to be transferred from a higher temperature molecule to a lower temperature molecule?
You may be experiencing a very common misconception, which I blame on Star Trek and Star Wars and all their talk of pure energy that always comes as bright lights, with the implicit assumption that energy is some kind of fiery bright substance, like a glowing fluid or something. It isn't. Energy is a property of matter, and of the rather peculiar substance we call electromagnetic radiation, chunks of which are called photons. Think of energy as kind of like an adjective that describes a noun (some kind of matter, or radiation). An atom can be energetic or not energetic, and so can a photon, just as a cat can be furry or not furry. But you would never mistake "furriness" as some kind of object itself. In the same sense, you should not mistake energy as some kind of substance itself. It's a property of substances, just like furriness is a property of cats.
So energy has to be transferred from a hot substance to a cold one, and indeed it could go by way of photons. One particle could emit a photon, transferring part of its energy to the photon, and maybe slow done, and then this photon could be absorbed by another particle, which would receive the transferred energy, and maybe accelerate (go faster). That's definitely one way it could happen. But it doesn't have to happen that way, because energy can be transferred other ways. For example, the first particle could simply run into the second, bang, and the impact itself could transfer the energy directly, no photons involved. Or the first particle could run into a third particle, or the wall of a container, and the second particle could pick up the energy from the third particle, or the wall of a container. And so on.
To add a little to the answer by @Christopher Grayce, molecules have kinetic energy due to the fact that they move about and also internal kinetic energy because they have bonds that vibrate and some bonds can rotate as can whole molecules. For a photon to give up all of its energy to a molecule it has to be absorbed and in doing so the photon is destroyed and the molecule is left with this extra energy which, depending on just how much energy the photon had, can excite rotations, vibrations or form electronic excited states or all three. Because molecules can collide with one another this extra energy is eventually spread out among all molecules. (Some energy may also be lost as fluorescence from an excited state).
We see things as being coloured due to visible photons producing electronically excited states and feel heat from a fire or radiator because the molecules in our skin absorb infra-red photons which excite vibrations in these molecules.