The tantrums and behavior problems sound like food intolerances to me, and yes, you have to avoid those foods. My DD is turning 3 in a couple weeks. We have a lock on our pantry so that she can't steal food. She does it too. If someone leaves in apple on the counter, she will steal it, and hide under the table to eat it. I take it out of her hands, and tell her it isn't her "happy food". When I feed her the things she can have, I tell her that is her happy food. Because when she has the "bad' food, she's up all night screaming in pain, and getting eczema, etc. My son, who is 7, also has food intolerances. Last time we tested him on milk, he screamed all night long. I said, do you want to drink milk and do this every night, or would you rather not have the milk? And he said "no milk." He's very good about checking ingredients etc. I do find treats for them. As in, DD gets maple sugar candy (the only one I can find that she can have), or I made sesame candy (sesame and honey - easy to make).
At that age, with those responses to food, you DO need to control her diet. And friends and relatives should be told of the food issues as well. Tell them she has food intolerances, for lack of a better term for the mood problems. But give into her on other things - let her have choices of what to wear, books to read, etc. so that she feels that she has a choice in some things. So you're not controlling everything about her. If you give her more of those choices, then maybe she'll be better on the food issues.
You sound like you're dealing with a lot. And maybe you need some professional help as well.
Good luck.