A bigger turbo providing more boost (and hence more pressure) will allow the engine to generate more power, if the engine is capable of doing so. But, lag will be much much greater.
Often, it is better to use a smaller turbo, or even two smaller turbo's (either in series or parallel) with a different engine format. Most cars nowadays use relatively small turbo's for this reason.
Superchargers work in much the same way as turbos from a pressure point of view. The system is still pressure regulated (i.e. there should still be a pressure relief valve), although boost is proportional to engine RPM up to that limit. At higher revs they are indeed less efficient, and they also require more power to run - a turbo is driven by waste gas, a supercharger is driven by flywheel power....
With regards to the air heating with compression, it's a very real effect. The amount air heats up has to be seen to be appreciated. I've done some calculations in the past on a Starlet Turbo engine, and the turbo at peak boost will heat the air from ambient to about 150-200 degrees! The effect this has isn't so much on the density of air (although it is affected), but rather on the thermodynamic cycle. I'll dig out some graphs at some point over the weekend to explain what I mean, but basically it's beneficial to cool the compressed air. Hence an intercooler. But intercoolers massively increase the intake tracts, and have a massive effect on lag, so fitting a bigger intercooler for a given turbo size/engine performance will not necessarily increase power outputs.
It's a huge science, and thats why car manufacturers spend millions of pounds finding the right turbo (and ancilleries) for a given purpose. Someone who thinks they can bolt on, say, an HKS intercooler and expect it to work off the shelf is almost certainly deluded. Bigger isn't always better.
Ok OK fine that last bit is garbage....
illepall 
yeye.. stop flaming! 