No, in the assumptions there are some (very common, I think) misunderstandings.
The third and fourth is right, of course, this is very general.
The second assumption has nothing to do with quantization, you mean it should also work for macroscopic billard balls, right? Consider two counterexamples:
a moving ball frontally hits one that is at rest. Afterwards the first is at rest and the second moving, they have exchanged their energy. So indeed the less energetic one lost energy to the other -- but too much, the energies did not move a little to the mean value, but swapped past it.
a massive and slow ball gets hit from the back by another which has double velocity and half of the mass. Obviously the first one had more energy, and obviously it will get still faster and further increase its energy. So even this is possible (but only with differently massive particles though).
You are right in that sense, that if you have half of the particles with high energy and half of them with low, than after some time the mean energy of both kinds will become equal. Even it they were of completely different type. But every single particle will have a random energy, which can become arbitrarily high (with an arbitrarily low probability ;))
Your first assumption is a common misunderstanding. No, there is no minimum quantum of energy; only some cases where this seems so...:
Well, if you consider e.g. red light, than this is true, all energy values transferred will be a multiple of a certain quantum $\hbar \omega$. But that's because you chose to work with red light only, i.e. with particles of a certrain energy. It's not surprising, that they all have this energy :)
Also normally, quantum (= very small) systems have some energy levels ( = amounts of energy they are allowed to have), with gaps in between ( = not allowed). In that case the change of energy is obviously also restricted to certain amounts. But usually (not always, though) above a certain energy (which it is convenient to call zero) there are no more gaps and every energy is allowed. Those are called the free states of the system, as opposed to bound states with negative energy. Consistent to these notations, a free particle is in a free state and can have an arbitrary energy.
As to your last question: yes, if the particles were systems which are restricted to energies that are integer multiples of some minimal value, and if they interact only in a manner that allows energy transfers towards the mean value, then they would after a finite number of interactions all have the same energy (or with difference 1, if it doesnt add up).