Thanks everyone for the wide variety of responses, it's given me a lot to think about. I tried asking DD again last night why she doesn't like her school and she said that it was because not everyone spoke English and it was boring. Yet, she must have had a good day because she also said she had fun? It just baffles me at times. We do have an end of the semester parent teacher conference coming up at some point so hopefully we'll be able to find out more then. DH and I have both visited her school and I know there are a wide range of activities in the rooms one that I know she would enjoy but I think part of the problem is that she wants her friends to constantly do the work with her and while it's allowed many times her friends have different interests. Even at home she'll complain that she doesn't know how to play by herself and complains if she's supposed to do it at school too.
Part of the reason we are so worried about academic fit is that while there are a number of school options locally many of them only allow children to enter during the preschool age so we're somewhat forced to look at this long term (the one exception would be bilingual English schools because she's a native speaker, but most of them are well outside of our price range). Traditional schools (many of the "top schools" would fit into this category) tend to be very strict in terms of learning and definitely wouldn't allow children to work ahead and may even discourage children asking questions.
I do think a good part of this is simply her personality. DH and I have both struggled with perfectionism at various times of our lives and have certainly had very shy/socially awkward stages too (although my parents said that at this age they both don't remember perfectionism to the levels that DD has). I guess, this leaves me with the question... how do you strike the balance between pushing your child outside of their comfort zone to help them accept failure too but also not pushing too hard? I think this is something that we struggle with and just aren't sure how to handle.
In terms of true peers and socialization. That's a tough question and I'm not really sure if she does have true peers or not. She's one of the youngest in her 3-6 year old classroom and I know she talks to some of the older kids too but her two closest friends are 4&5 and I'm not sure where they are at schoolwise. I do think they have very different personalities than DD when I see them at school functions and parties. They appear to be able to go with the flow, run around and have fun and don't let things get to them. DD is just not that kid and seems to internalize and ruminate on everything. She'll dissect people's actions (sometimes even those of DH and myself) even when we had not specific intentions behind them. She also had a number of fears (heights, witches, even Disney characters) and those also seems to prevent her from relaxing as much.
We do talk a lot about conflict resolution, playing with her friends and how to deal with fights. I can't remember if it was in Nutureshock or not but I do remember reading that fighting in front of your kids can be healthy (as long as it's done in a reasonable manner and you come to some sort of compromise in the end). So DH and I make sure not to hide our rare disagreements but we discuss them in front of her and try to reach a solution so she can see first hand how it works. We talk about how to compromise with friends also (especially if we go to playdates).
I also don't remember ever saying she's "good" at anything but we make sure to say that she practiced a lot and that's how she was able to do XYZ. She certainly asks, though! She's constantly trying to compete with us and "win" (whether it be who can eat the fastest or an imaginary game she makes up).
I should say also that outside of school she does have a number of bilingual friends (both younger and older). She definitely seems less stressed in her friendships with younger kids (probably because she can dictate the relationship more) but still prefers those with older children even with the occasional fights (but among her friends outside of school the fights seem to be much less, although they're also around each other less too).
