What does "optically thick media" mean?
Does this property depend on the length of an enclosure?
Why is the diffusion approximation for the radiative transfer equation applied to optically thick media?
The diffusion approximation is one solution to the radiative transfer equation. In general, the choice of applying this particular solution depends on the optical limit, as you say.
For an optically thin medium, radiation will travel and may interact along the way. This is not characterized as a diffusive process, because the beam can interact with the medium anywhere along its path.
For an optically thick medium, radiation will be absorbed within a multiple of the mean free path. The initial volume element absorbing radiation may then re-emit radiation to its neighbor, which may in turn absorb and re-emit. This diffusive process continues into the medium. In the optically thin medium above, this process is not dominant and hence the process is not diffusive.
An optically thick medium is one for which the mean free path of a photon is low. This means that a photon won't be able to travel very far before it interacts with the matter than makes up the medium.
The measure of optical thickness, optical depth, does depend on the volume of material in the medium. For example, for a material with a fixed density, light travelling through 1 kilometer of this material will be less likely to scatter than light travelling through 2 kilometers of this material.
I am not familiar with the diffusion approximation and so cannot comment on your third question. People, feel free to add your answers to the third question to my answer.