Re: Why did the League of Shadows want to destroy a peaceful city?
It wasn't so much about Gotham itself this time (except that the peace was built on a lie, which doesn't mesh with the more purifying thoughts), LOS wasn't really LOS anymore. I'm not sure Ra's would have agreed with Bane's plan. It was more about both Bane and Talia's own personal vendettas and goals this time around. LOS was more of a tool and a cover for the true intentions.
Bane wanted several things. He wanted to prove to himself that he could do what Ra's failed to do, as a kind of response for him being thrown out of the LOS. He also wanted to extend his own misery and pain unto others, on a larger scale to Gotham and to a more personal level to Bruce. With both Gotham and Bruce he makes them suffer what he had to suffer, the false hope that he consider the worst suffering and misery of all. Bane is also in this for his love for Talia (whether it's platonic or not doesn't really matter) and wants to prove himself to her while also helping her get revenge on her father's murderer.
Sometimes I think people want to make TDKR simpler than what it really is.