Agreed with above. I just did a similar install (with the thin stuff over sheetrock) 2 days ago in a new home. I had a heck of a time explaining to my co-worker why his ideas about absolute plumbness were irrelevant- I still don't believe he understands. If the existing edges look plumb enough then you simply follow them. Even my experienced eye cannot see something a little out-of-plumb but a taper jumps out at you and will be seen, so that the idea is to avoid tapers moreso than obtaining perfect plumbness. Even in this brand-new home had we followed plumb we would have had a visible taper at both ends and in every corner!

The corners or ends are best handled by measuring the space which will be left after the last full width board is in place and deciding whether it needs to be split into half that on both ends or if it will look better in one piece. Also keep in mind before you start that you can change that a little shorter by running the adjoining wall first (so long as it doesn't cause the same issue there). This is all about aesthetics, and a little about practicality as you notch under windows (or on a full-length install around doors too). In those cases, you can often avoid an excessively narrow rip by shifting out of one corner a little (so long as it will still look good elsewhere). One thing not discussed here yet that I always do with these is to use a urethane glue (liquid nails is one) underneath the boards. Blocking at or under 16" centers will do for support, and I'd be sure it was tucked under the sheetrock on the wall above to support it's lower edge as well.

Firecode may insist there be sheetrock underneath. Practicality says that if a fire is so hot that it burns through 3/4" of wood then it wasn't going to be stopped for much longer by the 1/2" sheetrock on the wall above this. I do know that your insurer will care if you need to make a claim, and if an inspector is involved then it must be done by code. How you handle legalities is up to you, perhaps you're not pulling a permit for this and you'd claim that the wainscot was there when you purchased the house if the insurer didn't have pics or a written report saying otherwise. I wouldn't see this as a substantial hazard and that it's up to you to keep you legal, not me. All I can recommend here is that you know and follow current code where you're at- it's there to protect you.

Phil