If she goes to bed at 9.30 what time does she wake up? A late bed time and disturbed sleep seems like a recipe for tears and having trouble coping with other kids. I know some gifted kids don't seem to need much sleep but my experience is the "sleep begets sleep" adage is pretty accurate. She is only three so unless she is napping in the afternoon I would try changing to a 7.30 bedtime and see if it helps. You would need to be able to commit to it for at least a month though.