You "almost" pulled her out early? When she behaves that way, you leave. Immediately. I'd also stop all extra-curriculars for a while because she's clearly not ready for them.
You say that you've tried everything, but all you've really tried are punishments that are unrelated to the misbehaviors. And grounding her from playdates isn't really going to have the effect you want at this age since she won't necessarily tie her misbehavior to you saying no to a playdate 5 days later.
How does she behave at home when she's just with you? Does she listen? If not, you need to work on that first. When you need her attention, get on eye level, touch her shoulder if necessary, and say her name. When she looks at you, tell her what you need her to do. You can even make tasks into fun games, but the important thing is that she's doing what you need/want her to at that moment.
Before you go out in public, remind her of the rules -- things like staying with you, walking (not running), using an indoor voice, etc. If she misbehaves in public, you can give her a warning for something minor. If she continues or if she starts a tantrum, you immediately go home. Immediately. And you make sure that home is not as fun -- no TV, no playing with friends, etc.
The main key is that you can't let even a single misbehavior go. She has learned that she can do whatever she wants because you won't stop her. And think about logical or natural consequences whenever possible. If she hurts someone's feelings, she needs to apologize or write a sorry note (even if she can't really write yet). If she throws a toy in the house, that toy goes in time out. If she runs away from you in the parking lot, then she needs to hold your hand or ride in the cart/stroller until she can stay close.
You also need to make sure that she's receiving lots of positive attention from you. Play with her, snuggle her, read stories with her, etc. When she realizes that good behavior results in good attention, she won't want to misbehave.
One final thought... her behavior honestly sounds more like a 2-3 year-old than an almost 5-year-old. You may want to consider talking to her doctor or seeking counseling if you think her behavior is medically based.