I would think that he would be able to understand her. Edward wasn't always able to read her thoughts, it was when Nessie was further along in development, that he started to hear her thoughts. Given what we know of her character she was very advanced, ie reading Tennyson, and being able to talk early on.
As for Edward not being around, I think he was, but we see things only in Bella's POV after Nessie is born, so that is likely the reason why we saw more of Bella with Nes, instead of Edward and I don't think there was a question of him loving her after she was born, once he realized that Nessie loved Bella and in turn him, and after she was born he was quite taken with her.
"Tell me about her."
Edward "She's like nothing else in the world," he told me, and the sound of an almost religious devotion was there again in his voice.
I won't go through and quote all the parts but I will add this one from Renesmee's Memories. ...Or the song Edward had crooned to her that was so perfect Renesmee played it for me twice...
I like them as a family, and the happy ending, but anyway that's my input of sorts.