Paul, underexposing does not in itself cause noise; only trying to correct that in postprocessing does (by amplifying shadow noise in underexposed areas). Maybe it's a semantic difference, but I feel it's an important one.
I am not sure what you mean about "shooting to the right incorrectly exposes an image". If you shoot to the right taking care to not blow out any highlights, you are maximally using (getting the highest possible signal to noise ratio from) your digital sensor. You can of course decrease the exposure in postprocessing a bit, without adding any noise.
In general, sharpness is not a function of exposure, within a wide range of reasonable parameters.
Since you have to correct 'underexposure' then its a cause of noise in an edited image. Its actual effect varies depending on how much an image has to be adjusted and which tones need the adjustment.
ETTR (Exposing to the right) introduces tonal discrepancies in my experience - its not a solution but is touted as 'data acquisition maximisation' (and supposedly produces a more malleable digital file). Its simply not that simple in my experience - I've tried innumerable methods of exposing and its best to exposure for the image you want rather than apparently maximising data capture.
'Sharpness' is a function of many variables, exposure is just one. It doesn't matter much, except when all the others are additive.
FWIW the camera I have found to produce surprisingly 'malleable' files and respond least badly to hauling up shadows has been the Leica M9 with its full frame CCD sensor. Not usable underwater and 18MPixel only but surprisingly effective. I find the Sony A7II can be good but then again it can be disappointing - I suspect this is down to tonality problems which I don't always find fixable in post processing. As always depends what you are doing - I have been printing to 30" x 20" for the last couple of years, which is a reasonable size, and discrepancies can start to show.