ETA Based on your SWH, I think there may be something else going on. Have you discussed it with the pediatrician or the schools to see what early intervention services she is entitled to?
Original answer: The reason kids start with a tantrum is that they are frustrated about something. The reason they continue with them is that they get a payoff from others around them. So, for example, if she falls and starts wailing when not really hurt, if you fawn over her and soothe her constantly, she learns to get attention by wailing. Same thing if she wants a toy that someone else has - if you intervene and pay her off with a different (and better) toy, she'll continue to throw tantrums.
The thing to do is ignore the unacceptable behavior. A simple "I can't talk to you when you are screaming" or "I don't answer to yelling." Then walk away. Really. No matter how frustrated you get. And your DH has to get on board with this! He doesn't get to just leave with the other kid because she's a pain. That adds Daddy Rejection to her list of slights.
If she says she is never playing with him again, that's fine. But none of you need to listen to it. Send her off to her room with as little talking as possible, or go somewhere else with your son until she stops crying. She can go to her room to get control of herself, and (important) she needs to be welcomed back when she is calm. Meanwhile, you and Daddy and brother have fun playing a game. The game is for quiet people, for cooperative people.
I know you are probably yelling at her or trying to talk her out of her drama - but that's fueling it. You and your husband have to be completely emotionless about it, firm in your resolve. She needs to be taken out of places where she has a tantrum - stores, play dates, parks, Grandma's house. Those places are for well behaved kids who don't scream. She goes in the car, with no conversation, and you go home. NO MATTER WHAT. No going home to toys and TV or special time with you, but home to nothing. If my son had a tantrum in the car, I pulled over, got out, stood where he could see me (no abandonment fears) but where I couldn't hear him. If you all go off together and she starts with her nonsense, pull over in a safe place and you ALL get out for ice cream or to play a game on the phone, anything. She sits and screams to no one. The car doesn't start until the yelling stops. You'd be surprised how boring a car seat gets when there's nothing to do and no one to listen to. Even little kids understand that "tantrum + screaming = nothing" if it's consistent. Don't give up after the first or third or tenth time. It WILL work if you are the Rock of Gibraltar on this and cannot be moved.
And I think you have to look at the sleep issue. If she can't settle down in bed or in social relationships, you may need more help here.