but a couple of questions:
Does he really need a haircut? I know you say he has thick curly hair, but don't you think if you let it grow out, he will tell you when he is ready to cut it? My son has thick curly hair as well, and he didn't ask for a haircut until he was 4. He is 10 now and likes to grow it out and then cut it all off. Also, it might be good for him to watch you get your haircut, just keep reassuring him that you are perfectly safe and that everything is going to be ok. Let him watch it to the end so he knows you are telling the truth.
Does he really need his hair to be washed? If his fear of the bath roots from hair washing, maybe don't do it for a little to make him more comfortable with the bath (so that at least happens). Over time maybe you can talk to him about how dirty his hair is probably getting and ask him if he wants it washed. If he says no, then don't do it.
I also have to ask, although it might not go over well, does he have to go to the doctor? I understand if he does, but just think about it.
Ultimately, I would say that instead of coaxing him, include him and respect him when he doesn't want to do it. By talking to him about it over time, he will gain your trust (that you won't make him do it when it's clear that he doesn't want to), then he will become more reasonable about it.
Of course, unless something traumatic has actually happened to him that is making him act these ways, but it didn't seem like there was.

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Have you looked into him having some sensory issues, these are common fears of my guy, esp. the hair cuts, baths and hair washing. The doc place, I think its just what you said, shots an stuff. But the other stuff sounds like what my DS has had issues with check and see if your library has The Out of Sync Child. Of course he could just be extra scared too...I don't want to just jump in an diagnose your kiddo, just some food for thought.
My son is 4 and every bath is a struggle, so we only bathe like 1-2 times a week, hair washing is once every two weeks, and its still a struggle that ends in tears and vomit...good luck feel free to PM if you need some support.
"Have a safe lunch, use a condiment."