OP - all the advice in the world (may it be the best or worst) won't help you unless you decide to be an adult. It is extremely difficult and you are drained to the point of not giving a f.... That is expectable and "normal" with 3 kids this small. But it doesn't change anything. In order to protect your kids (all of them!) you need to be strict. Even to the point of actually not caring. Let me elaborate on that. Tantrums and behaviour in general is a power play. Any situation that gets out of hand is mainly because the players react on an emotional level and get drawn in. And that's when you raise your voice,...etc.
By not caring I mean that you are able to stay cool and keep in mind that you are not deliberately hurting you son in any way, you are just establishing control and with that helping him to be a kid and enjoy life.
Also being able to detach for the sake of a "greater good" and still be able to feel everything (love, sadness,....) takes practice. It is way too easy to swing the other way and get too detached.
If you can get to a point where you honestly don't care about the power play and are able to say: I am in control, these and these are the rules, then you can start changing things.
Wants to hold his sister's hand when walking. You can if you don't hurt her. He hurts her: short leash (really awful, but sometimes needed as you only have 2 hands) or hold your hand and walk behind the pram with you. Result: tantrum. Fine. Hold his hand and drag him along. Keep going wherever you were headed. Don't go home - that means he won, he has the power. Don't start arguing: he won. Don't be fussed, angry or ashamed - he won.
Just keep going.
Signal: I am in control and unless you meet minimum requirements of behaviour I will just keep on going/doing my stuff until you come around.
The other thing you will have to let go of: what others think in public. Just forget it. They are not in your shoes, they don't know why your doing stuff and a few minutes screaming won't hurt anyone.
Burger king: you were the one who let him play with the door. He is 4.5, you can still pick him up and physically restrain him. You are his mum. Sit him on your lap when he's eating, otherwise hold his hand while waiting. He doesn't get to go around and cause pain. discomfort/disturbance to other people just because he is bored, feeling angry,.... That's not how societies work and not how families work.
Again: the result will be a tantrum. If you already have the food: Offer him the option to sit in your lap and eat, or to leave the food, go home and only have dinner x hours later. You will most probably have to do the latter. He is small, but he'll survive.
And even before that: whilst waiting in line he will tug and try to get away. No. Hold his arm and don't let go. He will scream. Fine, let him. Point out to him how many other people are doing it (hopefully 0). Show him that there is another way. Go down to his level and ask him to recite a rhyme or something that he needs to actually think about.
Or simply state that if he continues you will go and he will have to wait until dinner to eat anything.
Kitchen: I'm guessing the scenario, correct me where I'm wrong. You are at the counter, doing something, 2 year old is around your legs, somewhere/roaming or let's say playing. Smallest is in pram, high chair, cot? DS comes in.
Before anything happens order (ask him very firmly!) him to come to you. If he won't, put down what's in your hand, hold his hand, take him to the bench (your previous position) and tell him you asked him to come to you and you expect to be listened to. At this point: you expect this because you are the adult and in control. (Sorry to say, but being a kid is tough). Then immediately give him something to do. E.g.: Go fetch a book from the living room/his room. Then ask firmly (order) him to come next to you (if possible on a small step so he can reach the counter) and put the book there and show you this or that. Just say you remembered something from this book and you need him (his help) to find out more. You can only remember it partially. It was something yellow....blabla... If he start to even look into sis direction instantly react to it: I've asked you help with this, could you focus on it?
Again: you control the situation and not him. Also you are teaching him to control his emotions and impulses, by controlling them for him.
Every child needs to be on the reacting side mostly, as they are not mature enough either mentally or emotionally to handle control.
Through all of this please don't forget to tell him you love him and would go to the moon and back, but his behaviour is not acceptable. You love him, but he (as everyone else) has to do the bare minimum in order for this family to work.
Unfortunately for you OP, there are millions of little tricks, but each situation needs a controlling behaviour on your part and if it's not in your nature then finding these will be hard. And there are millions of situations every day. It is tiring and noone is there to show you how these work. And noone can help you endure the tantrums that will come in the first few weeks.
Unless your DS really has an underlying issue, manning up, controlling the situations and being strict should have an effect quite soon.
And one more thing: You should pick your battles carefully.
I'd suggest the same as many have done: create daily routines: fixed times to get up, eating at the table for all kids,...
Once these are really routine and no kid challenges them (a week or so I'd guess) move on to the next one: harmful behaviour. Try preventing them with controlling the situation (and giving DS stuff to do).
I have to say the idea of replaying a bad situation and showing how it could have been done better is great!