Chickenwing, I think you've explained the general concept pretty well, but a couple of your statements need some refining. First, let me say I'm not an authority on this subject, and frankly, I don't know that anyone currently in our forum is either.
Anyone contemplating twincharging (compound or multistage compression) should do as much research as possible on the subject, to where you're comfortable with the concept, and the physics. As usual, there's a lot of hype out on the Internet about this subject, but the real truths can be found from talking with engineers who know how to size and design multistage air compressors.
A Roots blower by definition has no internal compression. It's the restriction downstream (the engine) that creates the "boost". Of the PD machines, only the twinscrew has internal compression. But that's immaterial, because "any" time you vary the pressure or volume of air with a compressor, blower, turbo, or turboexpander (the opposite of a turbo), you're going to heat or cool it.
The Roots blower, by design, isn't suitable for high pressure ratios because again, it's not really a compressor, it's a blower. That's why it's not used in high boost applications. The other type of PD charger, the twinscrew
is suitable for high pressure ratios (you see lots of twinscrew air compressors out in industry).
The equation of (P1V1)/T1 = (P2V2)/T2 is not actually the Ideal Gas Law. It's the Combined Gas Law, so called because it combines Boyle's, Charles', and a French guy with hyphenated name, who I can't remember, three laws. Most of us here probably don't care whether it's the Ideal or the Combined, but if you're going to quote it, then it probably ought to be correct.
Anytime you put two stages of compression in series, for compounding you have the possibility that one will overpower the other, so you have to run the numbers with their capacities to ensure it doesn't happen. Of particular importance (IMO) is to control the interstage pressure ratios to ensure they stay within the "happy" band of the technology you're using, and the temps stay under control. (You don't want to let your turbo get into surge, or you'll wreck it, neither do you want to be trying to create 3 bar of boost to the engine with a Roots blower, because its discharge temps will be way too high.)
Haven't looked closely at your table, but it's pretty easy to figure basic compound compression, if you assume no losses and no heating
(which is not a real world situation) You just take each stage's pressure ratio and multiply them together. Note that the pressure ratios are MULTIPLIED, not ADDED. For example if your boost in the first stage (turbo) is 10psi, the pressure ratio of this stage would be (10+14.7)/14.7 = 1.68, and if your boost from the 2nd stage (supercharger) is 5psi, the pressure ratio of this stage would be (5+14.7)/14.7 = 1.34. The combined pressure ratio would then be 1.68*1.34 = 2.25, and converted to "boost", it would be 2.25 = (x+14.7)/14.7 ; solving for x would give you 18 psi of boost.