As for Nathan in Sylar (though it may be a moot point now) trying to find out what happened in the death of his old girlfriend and then confessing to her mother, and that being out of character for both Nathan and Sylar: I think this "Nathan" was more Nathan as Parkman saw him, than the true Nathan.
Parkman didn't have access to all, or even necessarily most, of Nathan's memories. He mentioned to Mrs. Petrelli there would be holes, and she countered that there were enough objects for him to read to fill in the gaps. Between the missing pieces and Parkman's own perceptions, I think the result was a kindler, gentler Nathan. More emphasis on the hero, the statesman, and the father coupled with no memories of the more questionable or ruthless things he had done. It (seemed) to be more of Parkman creating Nathan in Sylar's head than a direct brain transplant. (If Nathan had still been alive at that point to pull from, I wouldn't be surprised if Parkman could do some sort of transplant. Or at least create something much closer to the original.)
As for why Mama P. gave "Nathan" the baseball hat in the first place, I doubt her first association with it was his girlfriend's death. It looked like a box of stuff from when Nathan was younger. "Nathan" was struggling with who we was and it made sense to give him a bunch of stuff likely to trigger happier memories (like the plane did, and that trophy sticking out of the box likely would have.)