I have no actual information regarding the CS/CSS manual gearbox ratios but given that they use the same engine as the R, I would imagine it's more possible to have the R's gearing as well, as opposed to the GTI PP but that's just speculation. Would be interesting if someone came up with the actual data.
Regarding the rest of your message, CS/CSS/R turbo is the IS38 as opposed to IS20 in the GTI PP though there are differences on the engine internals as well and not just a bigger turbo between them: Different pistons, cylinder head, exhaust valve seat rings, exhaust camshaft, piston cooling jets, high pressure injectors, high performance main radiator.
I guess it's also down to those differences that these engines can see up to 500bhp on stock internals. A stage 1 with around 360-370bhp and no hardware upgrades at all is perfectly fine from stress and cooling point of views. To answer mkviken's point about the oil temp, the EA888 gen3 has a much more advanced oil management system and those temps are normal, no point comparing to previous gens. The car tends to stay within a smaller temp range no matter the driving conditions. My 2011 1.4 TSI motor would sit at 90, climb as high as 115 after hard use and in general fluctuate a lot. The Clubsport sits at around 105 under fully normal use and only goes up 5 degrees at most. Even on track it never went more than 118 iirc. There are no cooling worries until after stage 2 or if you are doing frequent track driving.
Also, the chassis (speaking of the standard Clubsport here) is good enough to handle the extra power no problem, at least for a stage 1 I can testify that it doesn't ruin anything in the dynamics of the car, unless of course for the fact that the first gear is unusable and 2nd still needs caution which are worries if your do drag races but not on the track or public road. But the diff is a beast and it doesn't sweat at all with the extra torque.
I bought the CS as our family driver and the car I'll take to the track a few times per year and it hasn't disappointed in either role.
PS. Only because you mentioned something about leather not gripping you well in the corners: Go for one with the Recaro's, they're an amazing seat. Top grip and most comfortable seat I've sat on since a very long time, don't ask me how it's possible.
