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1) According to Grant Morrison DC has decided to let him treat Bruce Wayne as an immortal Batgod that can never get old nor actually die. If Bruce Wayne ever dies he just gets reincarnated in a new identical body.
Morrison didn't say Batman is now an immortal god. He's talking about the character concept, not the character.
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2) It is Wayne's old body, which doesn't really matter as Wayne is now an immortal Batgod that can just get a new identical body.
It's not Bruce's body. When Nekron said his name, we saw quotes surrounding Bruce's name. Whenever you see dialogue that doesn't use quotes, like in a comic, but you see quotation marks, it is either a quote from someone else or it is a reference to a false identity. In other words, that is not Bruce Wayne's body. But a body dressed in a Batman suit, designed to elicit an emotional response on the heroes who had died and were resurrcted, in order to turn them into Black Lanterns.