Class-A,AB,B, and C are transconductance amplifiers and included in linear range PAs. Among these amplifiers Class-C has highest efficiency but lowest linearity. There is a trade-off between linearity and efficiency. For high efficiency you can choose switch-mode amplifiers which are unlinear basically, such as class-D,E and S. I guess since Class-D does not employ the parasitic drain-source(collector-emmiter) capacitance, it is not suitable for high frequencies. Class-E is more efficient and simpler comparing to class-D. If you want to amplify a amplitude-modulated signal, you can use "envelope elimination and restortion technique" in a class-E PA.
For your second question, class-AB are biased to pass a part of signal less than 360 degrees (as linear class A do) and higher that 180 degrees (as class-B do).So this class is more linear than class-B and more efficint than class-A. the output signal then is filtered using series RLC filter to obtain clear sine wave.