1) Is this even possible, or will the inherent nature of the battery-based adjustable voltage power supply prevent it?
It would be a LOT simpler to see if your CPAP machine is happy on 20v.
You could connect two of those things in series, but you MUST NOT make any other connections to them while they're connected in this way - if you plug them both in to charge, you may short one and cause a fire.
Internally, they have a boost converter that steps the low battery voltage (~3.7v) up to whatever output voltage you select. You can connect boosters in series as long as they're isolated from one another, have similar current specs, and a load that doesn't pull their max current.
2) Is this dangerous to me? Keep in mind that this is powering a CPAP machine, so I will be always be asleep when using this setup, and therefore I am unable to monitor it for problems.
There's always a slight chance that anything with lithium batteries will randomly catch fire and explode. There's also a slight chance that the booster will suddenly die.
Since CPAP machines typically are to improve sleep and are not designed to be life support, I think you'll be fine if you put them in a protected area outside your tent.
3) Is this dangerous to the device? I.E. can the electronics deal with "24 volts flowing through it"? Or is that not "how it works"?
Voltage doesn't flow, it pushes. Current flows.
As stated before, you must not hook up any load that pulls the maximum output current of the boosters, or things may get wonky.
Note that the maximum current at 20v will be a lot lower than the maximum current at, say, 9v.
4) Other things that I don't enough know enough to even about to ask about.
It would be a LOT simpler and probably safer to obtain a 24v battery, or a similar pack that has a 24v output option.
You never told us how much current your CPAP machine pulls, if it's low to moderate then you could probably get a 6S R/C lipo (eg from hobbyking) that would last out the night.
Another option is to use a small 12-24v boost converter, with your pack set to 12v. This will allow you to hook both packs in parallel, although one will drain a lot faster than the other due to the boost converters being set for very slightly different voltages.