It sounds to me like she is just not tired. Some children require less sleep than others. Also, she may need more physical activity during the day.
I know you say no nap during the day ,howeveer If she is in daycare very often they do have sleep periods find out how long they nap for Also make sure she is getting enough physical exercise, playing ,stimulation, and weather permitting outside activities, parks having Fun .No big eating or drinking before bed the sugars in Milk can keep her awake ,have a cut off time ...Actually another thought does it matter if she sings to her self and pick her nose, how does it affect you all, if she is in her own room and in bed that seems okay ..Good Luck