It sounds like there needs to be some improvement on both sides, from the teacher as well as from you and your son.
What your son has been sent to the headmaster for sounds very small but as a teacher I wonder if they were the last of a series of other small misbehaviors. If he is being a distraction, it is best to get him out of the classroom for a bit. It may be that the teacher is now watching your son more closely because she expects poor behavior out of him. He has set the pattern. This can be reversed with some very timely actions on your son's part. Have him start complimenting his teacher. Ask him what she does that a good teacher does. For example, tell him to tell her that he likes how beautifully she writes. That way your son will start looking for things that he may like about her instead of telling himself over and over that she is evil and doesn't like him. Let him know that you will be asking. If he changes his mindset, it will have a better effect on his behavior. And if she sees that he is not just looking for ways to annoy her when he gets bored, that will help her to deal with him better.
Ask the teacher what your son is doing well with. Let her know that you just want to build up his self-esteem. She will probably realize that she needs to be praising him as well without even being told to. Then tell your son what the teacher said. It will be good for him to know that the teacher notices his good behavior or his academic ability, etc.
Get to know your son's teacher. Volunteer in the classroom. It benefits your son when the teacher sees that you want to be helpful instead of accusatory. It sounds like your son and the teacher don't have the best personality match but that doesn't mean you should give up.
Then stop believing that your son is giving you 100% of the story. In your number one, it sounds like he spit his applesauce at the girl. You don't have to blow that hard to get applesauce off your lip - he could even use a napkin. But for it to land on the person opposite, he would have had to spit pretty hard. I would have scolded my son for spitting it instead of using a napkin instead of siding with him and then criticizing the teacher. I hope you did not respond to your son by criticizing the teacher. to him.
In number two, he should know by now at age 9 that if he isn't sure about an assignment, he should ask (politely). I would have scolded my son for not doing the paper as asked. And if he didn't think he was supposed to repeat it, then to simply ask for clarification.
In number three, I hope you told your son that you expect him to raise his hand when he is expected to. If you told your son that you can't believe he was reprimanded for something so small, he is learning that he can be disobedient and you will excuse it.
About number four, I would ask you if he has ever rolled his eyes at you. If he has, and he had misbehaved to the teacher, then he has basically given you every reason to believe he COULD HAVE rolled his eyes at the teacher. But your son has learned that all he has to do is swear he didn't do it and you will believe him. Think about it, who do you think would lie about him rolling his eyes - the teacher or a 9-year-old who gets fidgety, daydreams, doesn't like his teacher, and has a mom who will believe him over the teacher?
I'm sure daydreaming for a few minutes isn't really a problem unless he has stopped doing what he was supposed to be doing after being told repeatedly to get back on task. As a teacher, I can excuse a little daydreaming here and there, but if it is a chosen behavior to get out of doing what he is supposed to be doing, then it has to be addressed.
Your son should not be excused from standing up in front of the class because he is shy. Teachers know which kids are shy and will work with them to help them through it. One way is to have everyone take turns going in front of the class. When he refused, again, I believe you should have told him how disappointed you were that he would disobey his teacher. Then next time have him tell you when he had to get up in front of the class. Let him know that you can talk about how it felt, what was hard about it, how it went, who else was shy, who isn't shy, etc. By talking about it, you can assure him of when you had to do something that you were shy about and it turned out okay. You get the idea. Your son cannot do what HE wants to do when HE wants to do it and know that his mom will stick up for him.
Before I would change schools, I would stop giving your son an excuse for every bad behavior. Acting like all the other boys is not a good reason to allow bad behavior. And if he is punished at school for something, I would add to the punishment at home. I would start rewarding him on a chart for good behavior. For example, every day that he didn't get sent to the headmaster, he could get a sticker. Also praise him and tell him how proud you are of him. Each sticker would mean he gets to play with something he likes to do everyday when he gets home. 3 stickers gives him 20 or 30 minutes with you of playing a game, reading, etc. (his choice). Rewards are most effective if it is non-monetary and really effective if it is one-on-one time with a parent. 7 stickers would mean a trip to the park, library, museum, etc. You decide the number of stickers - I was just giving ideas.
If his behavior persists, then a couple meetings with a psychologist might help to find out if there is something going on his life that feels out of his control and is making it hard to behave. Sometimes kids won't open up to us because we might not like them anymore or we might overreact.
Also, have him start inviting a friend over from his class. After a few tries you will be able to see who enjoys your son and who he likes. Don't take it personally if someone doesn't want to come. They just might not have enough in common. Keep trying.