temperature - What is the difference between gas and dust in astronomy?

In astronomy, there is no formal definition of the threshold between gas and dust. Gas can be monoatomic, diatomic, or molecular (or made of photons, in principle). Molecules can be very large, and in principle, dust particles are just very large molecules. I've seen various authors use various definitions, ranging from $sim100$ to $sim1000$ atoms.

This is not to say that there isn't a distinct difference between molecules and dust. They have very different properties, but the transition between them is just not perfectly well-defined.

Gas, molecules, and dust can all be hot or cold, but if it becomes too hot, larger particles are destroyed in collisions. So while a molecular cloud typically is very cold and consists of both gas and dust, dust tends to be destroyed (though not completely) in the $mathrm{H,II}$ regions around hot stars through collisions with other grains, sputtering due to collisions with ions, sublimation or evaporation, or even explosions due to ultraviolet radiation (see e.g. Greenberg 1976).

To answer your final question, I haven't heard the term "gas" used for dust particles, but metals($^dagger$) and molecules can both be referred to as gas. For instance, $mathrm{Mg,II}$ gas is routinely used to detect distant galaxies, and molecular clouds contain $mathrm{H}_2$ and $mathrm{CO}$ gas.
In the interstellar medium, roughly half of the metals are in the gas phase, while the other half is in dust.

$^{^dagger}$^{"Metals" in the astronomical sense, i.e. all other elements than hydrogen and helium.}