Short answer to both: yes, and yes.
Medium answer to #1, you can use any source of high voltage as long as it is DC and the correct ballast resistor is attached to the tube anode to limit tube current to 4-6mA (for a typical, garden variety 2-5mW 632.8nm tube). You won't want much over 10,000 volts at the final output. This is enough to start any tube, and once the tube starts lasing, the voltage will be pulled down to anywhere from 2-4KV. HeNe tubes are negative resistance devices, so a ballast resistor of... oh... 75K with at least 5 watts power handling capability is an absolute necessity. Without it, the tube will fire intermittently, sputter, and overheat.
You will need to keep the capacitance between the ballasted power supply output and the tube as low as practical; the easiest and most important means of doing this is to mount & connect the ballast resistor as close to the tube anode as you can easily get; certainly not more than 6" if it can at all be helped. Any more, and your tube may fire intermittently or "sputter" and this isn't good for the mirrors.
Long answer, a trip to the HeNe power supply section of Sam's Laser Faq (
http://www.repairfaq.org/sam/laserhpd.htm)should provide you with a bountifully overflowing toi - er - cornucopia of information regarding homebrew laser power supplies.
For #2, a stun gun can be used to initiate plasma ionization as long as you do not allow the stun gun's arc to travel to either electrode of the HeNe tube, as that would put the power supply unit itself at risk, especially in a floating ground or ungrounded system. Most common stun guns produce a very fast rising spike of greater than 50-75KV, and would exceed the maximum voltage ratings of virtually all other components in the laser's HV power supply, letting out their entire supply of magic smoke in just a handful of milliseconds.
Best way to do this is to wire it like you would a strobe light's trigger electrode: a bit of aluminum foil or thin wire wrapped around the glass part of the tube between the anode and the capillary (try to wind this wire or apply the foil patch at least an inch away from the anode connection itself); and run ONE wire from this to ONE probe on your stun gun. If you don't see the end of the tube dimly flash an orangish color when you fire the gun, put the wire on the gun's other probe and try again. Be sure your setup is grounded, and sit (with feet off the floor) or stand on a plastic lawn chair or plastic step stool just in case the main power supply arc propagates in all the wrong places and ends up going up the gun and out through you.
On a battery-only supply, you need not go through the chair hassle because there is no return path from you back to the power supply.
SAFETY TIP: Hold one hand behind your back when firing off the stun gun or messing with any part of the power supply while it is live. This effectively prevents you from becoming part of the circuit, so long as the rest of you is insulated from ground.
While having an arc propagate through the system and into you through a stun gun is
SO VERY UNLIKELY, you must not overlook strange things happening when you mess with high voltage.
NOTE: This is *not* a recommended way to trigger a HeNe tube; as most do not require an external trigger. But as a quick and dirty fix to bring temporary life to an ailing tube or a sucky power supply, it oughta work if the anode voltage is high enough to sustain a discharge in the tube once you get it going. HOWEVER... the Sam's faq does included several pulse starters that will do what you want your stun gun to do; so it's not entirely uncommon to need a starter.
