A couple of things.
Yes, you could have a THC directly drive your Z axis, with the right code. The basics would be pretty easy, but there are subtleties that would take some tweaking. This could probably be handled by a single Ardunio, depending on your stepper or servo drives. Or you could split the workload between two if you are doing all the low level motion control in software.
Depending on how you are generating your g code, you might want something with a little better front end like Mach or LinuxCNC to give you a nice MDI, preview and other features that are good for hobby machines. Even something like a Raspberry Pi can make a good front end to an Arduino motion control solution. Then you have enough memory and processing power to work with more complex files and be able to keep up.
Voltage to a THC is typically fed from a voltage divider, either built into the plasma unit or by an addon board installed inside.
True pilot arc is somewhat built in to the design, not that easy to add. Starting can be blowback, high frequency (not recommended for CNC), high voltage, etc. This is somewhat easier to make changes to, if you want to go from one to another.
But the big issue is that low cost plasma cutters and CNC do not mix. The duty cycle is way too low and they die in no time. So it's not worth investing any time in improving your unit. If you want to use it to build your CNC and THC and get all the bugs out before buying a good plasma unit, that's ok. You don't have to worry as much about blowing up something expensive while you are learning. But just don't expect it to do the job for a CNC cutter. Many of those type units don't even hold up to extended hand use, they are really designed for occasional light duty use, and anything more than that takes them out in short order. Built down to a price.
Long arc, short arc, heliarc and in-the-dark!