Can any gas be turned into plasma?
Yes, in principle, any element can form a plasma. For instance, the sun's outer atmosphere, called the corona, contains iron in multiple charge states as a plasma constituent.
Can multiple types of gases together be turned into plasma at one time?
Yes, again the solar corona is a perfect example of this. It is comprised of hydrogen, helium (both singly and doubly ionized states), carbon, nitrogen, oxygen, iron, etc. The latter heavy ions are found in multiple ionization states.
Can plasmas be turned into another state of matter?
Yes, if you reduce the energy of the system, the matter will eventually undergo recombination to form neutral particles without further external work done on the system. From that point, the normal phase transitions expected from thermodynamics are expected.
In a plasma state, do like molecules attract?
No, not in general. In a plasma there is a principle called quasi-neutrality whereby the charged particle fields act to cancel each other out over distances larger than the Debye length. Thus, two fully ionized iron nuclei are not likely to attract each other but rather repel each other due to their large electrostatic potentials.
Do plasmas of different elements have different densities?
I am not entirely sure what you are asking but I think the answer is no and yes. The number density of a plasma is not necessarily determined by the constituent species but rather the boundary conditions of the system. The center of massive stars can be considered plasmas of heavy elements and are much denser than corresponding outer layers of lighter elements during the latter part of their lifetimes. Early in their life their cores are often more heavily populated by light elements like hydrogen and helium with extremely high number densities.
The mass density, however, will differ from one mass species to another for the same number density, of course.
In the plasma state, they are piped to another storage area where they are sorted into like elements based off density.
They would not be sorted by density like oil and water but rather more likely by mass-to-charge ratio using magnets or centrifuges.
If it is possible to change the state of matter for such waste products, is it then possible to take the resultant distilled elements and change them back into useable solids/liquids/gases?
If energy were not an issue, then yes, in principle, this would be a possible procedure for neutralizing and eliminating toxic waste. The problem is that many toxic waste products are composed of complex compounds which would first be dissociated prior to any ionization, which alone would require tremendous amounts of energy for any industrial-scale application. To then ionize the dissociated parts would require, in most cases, even more energy.
The typical energies here are hundreds to thousands of kJ/mol for dissociation and ionization. Industrial-scale applications would involve millions to billions of moles of toxic materials, thus billions to trillions of Joules just to get to the ionization stage (this is probably a weak lower bound). These estimates all ignore containment and efficiency of the dissociation and ionization processes (the latter two of which are almost certainly not 100%). My quick, hand-wavy guess would be that such a factory would consume power in the range of megawatts, i.e., the amount produced by a typical nuclear power plant.
In summary, unless the price of energy drops precipitously and the need for such a system becomes paramount, I doubt this is currently feasible.