Generally to be considered a plasma the plasma parameter – basically the number of particles in a Debye sphere – must be large and the electron plasma frequency should be large compared to the neutral-ion collision frequency (assumed to derive from elastic scattering, also called binary particle collisions). The latter constraint helps eliminate salt water, for instance, from being considered a plasma even though it contains ions in solution.
If the number of neutrals is large, then the plasma will be collisionally mediated much like Earth's ionosphere or the solar chromosphere and below. However, in the chromosphere most of the gas is ionized, thus mediated by Coulomb collisions. The distinction is that the interactions are considered long-range interactions instead of the "short" range interactions of binary collisions.
The only way they would be affected is through collisions that occur in the plasma. But wouldn't this break down at low densities?
If the gas stops exhibiting collective behaviors mediated by long-range interactions (i.e., Coulomb interactions), then it is no longer a plasma. This is not so much regulated by density as it is by relative densities between charged and neutral particles. Another way of saying this is that the Debye length be small compared to the size of the system. When the Debye length is small compared to the mean free path for collisions, then the plasma is strongly ionized and electromagnetic effects dominate over the typical hydrodynamic effects one considers in a neutral fluid.
Since a lower number of collisions would take place, the gas would stop behaving like a plasma. At approximately what pressures does this happen?
The solar wind has a mean free path for Coulomb collisions for a typical thermal proton of roughly one astronomical unit or ~1 AU. The Debye length in the solar wind is ~7-15 meters, on average, which is roughly a billion times smaller than 1 AU. The number density of charged particles in the solar wind is typically ~5-15 cm-3 and the thermal pressures are well below 1 nPa (i.e., less than 10-11 mbars or less than 10-15 atms). Despite the low densities and pressures, the solar wind is considered a perfectly valid collisionless plasma.
Say, I have a non neutral positively charged plasma with a charge density of 1 Coulumb/kg. With 70 percent of the plasma comprising of neutral gas. Would this behave just like a completely ionized plasma of the same charge density? (1 Coulumb/kg). What difference would there be?
The net effect of having a comparable neutral density is to inhibit collective behavior, i.e., electromagnetic effects. For instance, the neutrals would act like a resistive drag force to any relative drift between ions and electrons (i.e., currents). The degree of ionization for weakly ionized plasmas can be approximated using the Saha equation. To determine the actual behavior, one would need to simulate the equations of motion that include electron-ion, electron-electron, ion-ion, and charged-neutral particle collisions. There has been lots of work done on this for the Earth's ionosphere.