Do different plasmas from different elements have different properties?

Like is gold plasma more dense than iron's?

If you mean mass density, then for the same ion number density and charge state (assuming quasi-neutrality) the gold plasma would have a higher mass density than that of iron.

If you mean number density, then that would depend upon how the plasma was formed and under what conditions it currently exists (i.e., Is the plasma being compressed like in the core of a star?).

Remember, to be defined as a plasma one of the conditions for the ionized particles is that they should satisfy: $$ N_{D} = n_{o} \lambda_{De}^{3} \gg 1 $$ where $n_{o}$ is the plasma number density and $\lambda_{De}$ is the electron Debye length. Another condition that should hold on scales $\gg$ $\lambda_{De}$ is quasi-neutrality, which can be defined as: $$ n_{e} = \sum_{s} Z_{s} n_{s} $$ where $n_{e}$ is the electron number density, $n_{s}$ is the ion number density of species $s$, and $Z_{s}$ is the corresponding charge state (i.e., protons minus electrons for each ion atom).

So if you had two plasmas, one of singly ionized iron and one of singly ionized gold both with the same number density, $n_{o}$ = $n_{e}$ = $n_{i}$, then the gold plasma would have a larger mass density.

Is the iron plasma more reactive than gold?

I doubt this matters anymore. The typical energies required to break chemical bonds are much smaller than the energies required to liberate electrons from atoms. For instance the bond energy for $H_{2}$ ~ 432 kJ/mol and we know there are $6.022 \times 10^{23}$ molecules per mole. This corresponds to a dissociation energy of ~ 4 eV per $H_{2}$ molecule (1 kJ ~ $6.24 \times 10^{21}$ eV).

Meaning, as one adds energy to the element, it will likely reach a gaseous state of monatomic particles prior to the individual constiuents losing any electrons (i.e., ionizing). Once in a plasma state, the ions composing the plasma will no longer interact with short-range residual Coulomb forces as they do in chemical bonds. The ions in a plasma are now controlled by the long-range Coulomb forces produced by the interactions of their own electric fields with the combined electric field from all other constiuents of the plasma.

do they retain their chemical properties from more normal temperatures?

No, I doubt it (see my response the question about reactivity). At least in the sense that I think you mean.

There will be some properties that the particles would retain if they were not fully ionized (e.g., absorption and emission line spectra), but this again is probably not to what you refer.

The chemical properties of elements are controlled by their electrons. So the more electrons they lose, I am guessing the fewer chemical properties from their neutral state that remain.

The key properties of plasma that depend on the ion size are:

1. Electric conductivity since it depends on Coulomb scattering of electrons on ions and the rate of it scales as the ion charge squared.
2. Radiation properties - for fully ionized plasmas the bremsstralung (braking) radiation due to the Coulomb scattering of electrons on ions strongly grows with the ion charge; for partially ionized plasmas the line radiation may be dominant, also generally grows with the ion charge.
3. Thermodynamic properties, e.g., for fully ionized plasma with the ion charge Z the pressure is $p = n_i (T_i + Z T_e)$ where $n_{i}$ is the ion density, $T_{e,i}$ are the temperatures of the two species.
4. The plasma ability to affect plasma facing material surfaces certainly depends on the ion mass, heavier ions can easier destroy the wall material structure by knocking out atoms or ions from the wall surface (so-called physical sputtering)
5. Beyond that there is so-called chemical sputtering which covers sputtering by ions with energy below the physical sputtering threshold. What happens there is a [fully or partially] ionized ion approaching a material wall surface would pick up electrons from the wall surface, and then chemical interactions would start leading to formation of molecular ions etc.
6. Hydrogen plasma recycling on carbon wall is an example of chemical reactions playing an important role in plasma-material interactions: a hydrogen ion that enters carbon is trapped there because of chemical interaction (hydrogen bonds) with the lattice; however as soon as it picks up a second hydrogen ion they bond to each other forming a hydrogen molecule that does not interact strongly with the lattice, so the molecule can freely escape from the wall.

Yes. Plasma consists of ionized matter. Which contains cathode rays (electrons) which are indipendant of the matter and positive rays ( nucleii) which depen on the matter ionized. So as plasma is a mixture of both it shows a significant physical difference. Eg: Density, Charge/Mass ratio, Effect due to Electro-magnetic fields