My previous post was moved here so I did some backreading on this thread.
According to the book, and their personalities, I think Alice is actually more of the fun loving aunt of the two. She thinks of Renesmee, and Bella for that matter, as life size paper dolls to dress up. She only got the headaches while Bella was pregnant because she was trying to see Bella around the fetus, which was completely invisible to her, so it made Bella fuzzy. Like trying to watch a TV out of focus. I think in Eclipse it says Edward says that they think that the reason Alice can't see the werewolves is that there is some sort of "natural defense" they may be born with. That's only a theory though and probably wouldn't be the same reason that she can't see Renesmee.
I think that after Renesmee is born, she enjoys spending time with her as much as the rest of the family, though she may not covet her as much as Rosalie. I think Alice would be the one of the other family members that Nessie is closest to. Think of how Alice is with Bella. She's always looking out for her best interest sort of an older sister, but her idea of fun is always a bit more outlandish than the others. Rosalie may be a second mother to her, but Aunt Alice is the one you go to when you want to go shopping, if you want to do "girl stuff".
Imagine the households divided up, each couple living separate from the other. Edward and Bella are her parents, young but still protective and loving. Esme and Carlisle are the grandparents as well as Charlie, with the never ending learning that the old and young seem to gain from each other. Rose and Emmett, more into life for themselves. Rose with her vanity and Emmett because he seems a little thick and living in the moment.
Then there's Alice and Jasper, who are almost more "colorful" than any of the rest. Alice has no memories of being human, and enjoys being herself. Enjoys playing human as I think Renesmee would do as well. Especially when she's older. And because Renesmee was never human either, they share that perspective. Jasper's background is so different from the rest of the Cullens. I think she would be comfortable around him, not only for his ability to control emotions, but because Nessie actually seems to have some type of that ability also. She charms almost everyone she meets without even knowing that she's doing it. She also knows that she would prefer human blood. I believe Jasper would also find he enjoys her company because she is almost the opposite of him. Jasper had to fight for everything and though he had allegiences that attached him to some, he never felt love until Alice. Whereas, Renesmee has known almost nothing but love since she first came into the world. They have quite considerable differences, but sometimes those differences are what draw you to another person.
I believe all the households indulge her and cherish her because she is so unique and easy to love.
Of her parents, she'd definitely be a Daddy's girl. Even as old fashioned as Edward is, he wouldn't deny her anything that would truly make her happy. He may overreact to situations when she's older, but she still has him wrapped around her little finger.
Just my opinion though...