He's a very advanced child - he's into the terrible twos and the "three-nager" stage early!
I agree with saying a lot less and making the consequences HIS CHOICE vs. something you are inflicting on him. That's a subtle difference but an important one.
So, if eating is done in the booster seat, then when he gets out, he's telling you he's done eating. So, as Military Mom says, you let him scream but you don't try to fight the tantrum and get him to stop and eat. If he doesn't stop, and you can't stand the screaming, that's fine. Just put him in his room since he's a) not hungry and b) screaming, and tell him he can come out when he stops screaming. Remember that he will not starve if he doesn't eat a meal. Offer limited choices - he can sit in his booster in Seat A or Seat B (if it's convenient for you to move the chair) but he can't choose not to sit in it. So, as B says, fewer choices: Not "where do you want to sit" but "Do you want to sit at A or at B?"
I'd change up the biting thing - you are walking away, which is good, but I'd think again about saying "Mommy is not for eating." He's not eating you. And he knows it. He's biting you. Biting hurts and it's not allowed.
I agree about draining the tub when bathtime is over. If there's no water and he wants to sit there in the empty tub, fine. Find something else for you to do, like folding towels or scrubbing the sink/toilet so you can supervise without him thinking that you have all the time in the world. He'll get cold in about 3 minutes. If he screams constantly and you can't stand to be there, gather him up in a towel, half dry him, and put him in his room with a diaper and not much else to do. Say you'll come back when he's able to talk like a boy and not scream like an animal. It can't be a power struggle all the time, like Military Mom says.
If he doesn't get in his car seat, he doesn't get to go to the park or the play date. "We're not going because you're not in your seat." Period. Done. Back in the house and not for something fun. He needs to be bored because he didn't get in his seat. Offer him the choice: "Do you want to sit in the driver's seat or go to the park?" "Do you want to sit in your brother's seat or go for ice cream?" Then it's "Sorry, no ice cream because you picked sitting in your brother's seat." I always has to be about HIS choice, HIS decision to pick the unpleasant or unrewarding thing. That can make him more compliant while making him feel he has some control over the situation.
I agree about taking away the soap and preventing endless handwashing. I have mixed feelings about letting the water run though - it's a resource you don't want to teach him to waste. But you could consider something like plugging the sink and letting a certain amount of water accumulate in it. You could put a plastic cup in there and let him pour the standing water over his hands. But if he's going to make a huge mess and soak the bathroom, then that's not going to work for you. Otherwise, "Do you want to wash hands or read a book?" If he choose washing, fine. Then when it's time to read, the answer is, "Sorry, no time. You picked handwashing, not reading. Next time, maybe you will pick reading."
Just limit the choices, as others have said, and only give 2 choices that you can live with. The same thing works for you: "I picked quiet time in the other room instead of your screaming in this room."
Engage less, don't over-explain (they don't care), and let him see that, essentially, he chose to be miserable.