V.B.
Amy, you have just described my oldest to a tee. Something about those 1st born strong willed girls.... I have read every book "spirited child" etc. Some kids crave attention good or bad. When she is acting out calmly place her in time out, consistently every time. It will become boring. Dramatically praise her good behavior even if it seems silly "Wow, you did a great job not screaming at that store".
This too shall pass when her impulse control gets better.
Its hard not to want to give them the world and always keep them happy. However imagine her at 16 if she gets everything she wants now. Good luck to you.
One more piece of advice. KARATE!!! Seriously. We are at the manchester karate studio (off bridge street). Huge help with respect and focus and self esteem. Its really helped us.