I love D. D.'s term "Threenagers"! They are moody, they think of themselves!
This sounds so normal for a 3 year old. First of all, you just cannot ask a 3 year old "How was school?" In fact, you'll see a zillion Mamapedia answers that tell you not to both with that question for 5, 7 or 9 year olds either. You either get nothing, or "fine" or "I don't remember." What your 3 year old is thinking is not at all what you want to know! So at a quiet time, or when you're at dinner, or in the car, you can ask a much more open ended question like "What was the most fun thing you did today at school?" Or, better, look at the drawings or little projects he brings home (or what's on the bulletin board) and say, "Tell me about this. How did you make it?"
He's engaged in his own activities - totally normal for 3 years old. He doesn't understand the "manners" involved in responding to a greeting with his own greeting.
Twirling the hair? Not sure - it may be a habit, it may be a sensory thing, it may be an obsession. Time will tell.
Screeching? That's a tantrum, just like his other tantrums, and you don't give in to it. If bedtime is when he's overtired, then start a half hour earlier, and do it with quiet routines: bath, teeth, pajamas, story (not an action story), snuggle in bed with a lullaby or quiet talk, and lights out. If he's screaming, it's for attention - that doesn't mean you haven't given him enough. It means he's getting a payoff for screeching, which is additional attention even though it's negative. Maybe he doesn't want you talking to anyone else (like answering the buzzer), maybe he doesn't want an interruption to his routine, I don't know. But my screaming kid went right into quiet time in his room (none of this "sit on the step until you calm down and I'll sit here and keep you in your seat"). He needs to be deprived of your attention when he does this until he can calm down. I don't think it helps to try to talk over him or tell him why screeching is a bad idea. You just remove yourself from it and he will learn it doesn't work and that it's more frustrating than whatever precipitated it.
Try one of the good discipline books including the "spirited child" book - it's really important for you to have an in-control, calm attitude. Your stress, while understandable, can actually be making your child more anxious and unable to calm down.
If a tantrum has no benefit, it will stop. But that means you have to very calmly say, "I can't understand you when you are crying/screaming/storming off" and "I'll talk to you when you are done yelling and throwing things."
It also helps to give kids a warning time before they have to transition to the next activity. If you come in and say, "Clean up, we have to go to school" then he will feel interrupted and ticked off. If you set a timer or give a 2 minute warning - "2 more minutes and then we will clean up and go to school" or "Let's put your things over here so you can continue with them after we go to the store" you will start to get a better result. It doesn't happen the first time so don't give up - it's the consistency that's important. And don't get into the situation where you are bribing, like "If you clean up now, you can pick out some candy at the store." That gets old fast, and they wind up never doing anything without a return. And sometimes they just have to do what needs to be done.
It concerns me that you aren't enjoying your son. The more strong but loving structure and techniques you can employ with confidence and consistency, the better you will do with this child and with the second one when she gets into the act.