Could I use an electron gun to make plasma? Could I ionize a gas using an electron gun and knock off electrons in the gas to make plasma?
 A: Generally speaking, I think it is possible to do that.
I took a class about condensed matter physics, which includes several ways to produce vacuum. One way to make vacuum is to use ion pump after using a mechanic pump. The principle of ion pump is to ion the gas by an electron gun and then absorb the ionized gas by a electrode.
We can get two points from the process. First, as the vacuum can be made, the gas is supposed to be fully ionized, in other words, the gas will be turned into plasma if we don't absorb it. The second is, this conclusion suits the low gas pressure condition. For the normal gas pressure, I have no idea. Maybe it's hard to ionize that because of the high density.
At last, we should pay attention to the fact that plasma is quasi-neutral. As a consequence, ionize the gas by an electron gun must make the gas negative. From that aspect, it seems that the gun cannot produce a quasi-neutral plasma.
A: In principle you can. Electrons will colide with the gas atoms and ionize it, thus producing plasma. However the mean free path of an electron will depend heavily on the density, pressure and so on of the gas. In air at normal conditions electrons cannot go very far. In a low pressure gas however they do produce plasma in their paths.
A: Many plasmas are made in labs using basically an electron gun. You will need at least a rough vacuum (probably ~10-3 Torr) and a gas that is relatively easy to ionize. In practice, Argon is one of the easiest gases to ionize in a lab; Helium is also pretty common, and Hydrogen is supposed to be as well, although I would be nervous with the hydrogen. I doubt you will have much luck ionizing air at atmospheric pressure.
I made my own electron guns with tungsten filaments and metal screen to produce Argon and Helium plasmas in a vacuum chamber. We had a high vacuum (~10-7 Torr) and then fed in our gas to a pressure between 10-5 and 10-3 Torr before cranking up the current in the filaments.
