1. Isn't it too weird Batman and Bruce Wayne died at the same day?
They didn't, Bruce 'died' somewhere during the Bane attacks, not the last day per se.
2. How can anyone be Batman if everyone saw "real" Batman die?
Because Batman is a symbol, not a person. I would argue Blake is a back-up in case Gotham would ever need a Batman in an extreme situation. But the war has shown that Batman has finally inspired Gothamites to stand up for their city.
3. Isn't the "too lame even for Robin" scene of "Oh, boy, I'm a big boy now, Gordon told me there are no coincidence, so I'll say it to thugs and then act like there can be no coincidece" - feels more like a scene from "Batman and Robin"?
There are a few "sooo lame" scenes in the movie that I want to cry.
I don't understand what you're trying to say here.