Why only common collector method is used in class AB amplifiers?
Another name for that config is emitter follower. It provides current drive ability with voltage gain of (almost) 1. It basically just buffers the signal.

Nope. The semiconductors would blow up. Assume that each semiconductor junction (diodes and transistor B-E) has a voltage drop of 0.7V when forward biased. Try and simulate it.

It may not be the best, but most simplest and practical two transistor push pull configuration that is good enough for whatever the puropose is - such as learning the basics from textbooks. If you need better performance for another purpose then you can easily find better but more complex configurations that will fullfill the requirements of those other purposes.