So she CAN behave, she just won't with you.
She's asking for a ton of attention, and she's willing to take negative attention. Don't give it to her.
See if you can find a family counselor who will work with you and your husband to get you both on exactly the same page and help you develop a strategy of parenting that you can agree to and be consistent about. It's often easier to give in to a whiny child for the moment without realizing that it causes more trouble later on. You could also get a book from a source you respect - "How to Talk so Kids will Listen" is an oldie, and I think Dr. Phil may have something out there too. Use your library before you buy anything.
If she doesn't behave in the store, put her in the car strapped into her seat, and say the car isn't moving until she stops screaming. Get out of the car and stand outside if you have to. If she acts that way in a restaurant, take her home. Now. No third and fourth and eighth chances.
Get down at her level and look her in the eye, calmly saying "I can see that you are angry. I can see that you are upset." Get her to tell you what she is upset about if you can. She may not have the verbal skills yet but keep trying. Say "I love you. I will talk to you when you stop screaming and hitting." If she hits, protect yourselves and restrain her until she stops. Don't talk, don't yell, don't let her hurt you or herself.
If she doesn't want to eat, she doesn't eat. You cannot force her. Give her a select amount of time (she's 5, so maybe 5 minutes) and then let her get down. Use "All done? Okay, you may be excused." Send her away from the table and put away her food. If she is hungry later, THAT'S the food she gets. No treats, no rummaging through the fridge. She can eat or not eat, that's her decision. What to serve is YOUR decision. No child ever starved because of a tantrum. She needs to learn what she gets to control and what she doesn't. Don't get to the point of gagging and throwing up - that's an area within HER control and you cannot fight it, so do not even get engaged in that struggle. She may do it to the babysitter because she knows the sitter works for you. The schoolteachers she knows work for someone else and she's not trying to manipulate them to get at you.
Putting her in a corner may not be effective - you can't keep her there without physically restraining her, so it's within her control.
Finding out what is going on at night would be helpful. I would not put any child in bed after letting them fall asleep elsewhere as it CAN be frightening and it feels deceptive to them. She needs to learn to calm herself down, at night, in stores, at the table. That can't happen until you figure out what the triggers are.
And get your husband's company, or the pediatrician, to recommend a psychologist who DOES accept her coverage. I'm not sure why she has different coverage than you do - perhaps there is a disability involved that you haven't mentioned. In any case, whatever provider you have who accepts your coverage should be able to refer you appropriately. Be insistent and demand the services to which she is entitled. Use the school psychologist as a resource as well.