N., I think it's wise to look into this situation with the help of professionals right away. They can help you begin to help your son before his behavior becomes a deeply entrenched pattern and he develops a negative self image. Besides the testing for ADDH, and ODD, I would recommend seeking counseling with a child therapist to get to the bottom of his behavior, and to provide you with some new tools to use when discussing his behavior with him. Any diagnosis needs to be followed with counseling, and any behavior difficulties will benefit form it.
It's difficult to tell what is going on with your son without being there. I think you would have noticed ODD at home as well, and ADDH would have been mentioned by his day care. It sounds like he's reacting that way because he's overwhelmed, but why is he overwhelmed? When you find out you can make changes in his life, and teach him some more effective coping strategies.
Is the school day too long? Is he getting good sleep? Many children have undiagnosed sleep apnea which affects their behavior during the day, and is misdiagnosed as ADD. Has a new food been introduced into his diet at school (food sensitivities can alter behavior)? Is he impatient with waiting his turn in a large class, or impulsive, and there's too much pressure from the teacher to be self controlled. If he is used to a day care person who does not pay much attention to him (as you described above) and used to quietly doing activities on his own, a teacher's attention may be overwhelming to him. Maybe he is overly sensitive to the noises and movements of a large group of children.
Besides psychiatry you might look into occupational therapy for your son if there are any sensitivity issues and they can even help with ADD/ADDH.
As a teacher, I want to apologize for your son's teacher's comments. A child's behavior should NEVER elicit judgement or blame. I am so sorry she made those comments to you, they were not helpful. No matter how frustrated she was. It is a teacher's job (in concert with the parents) to work through a child's behavior challenges and to assist and encourage him to find effective ways to communicate and cooperate within a group setting. If extra resources are needed, they are available and teachers (and parents) are responsible to seek them out.
I have used a resource called CARE (through 4Cs) which will send a trained child development professional to visit (at least pre-school classrooms) to observe a child (in his classroom) who is having difficulty. the professional then works out a plan with the teacher and parents. CARE usually has a waiting list so it's good to find out right away if they can be of assistance to you and your son.
Finally, even though your son wants to learn numbers and letters, he may not be ready for kindergarten the way it's structured nowadays. It may just be a matter of waiting a year, when he is more mature. Boys often have a better school experience if they enter school at an older age. They mature in a different way than girls.
So if you decide that his overall readiness is the issue, you might find a pre-school that is academically based. That would meet his need for challenges, and his desire to learn letters and numbers, and yet provide a more flexible structure for him. And although tantrums, hitting other children, and hitting the teacher are not common nor acceptable behavior, they do occur more often in a pre-school than in a kindergarten. Preschool is a good place to work out these issues before entering public schools, where the expectations become higher.
Wherever you go, you want to find a teacher who will build a relationship with your son from the first day. Children yearn to be seen for who they are by the important people in their lives. When a child feels truly valued, he will have the desire (maybe not all the skills) to work WITH the teacher. He would do well with a teacher who would give him lots of positive attention, listens to him, "gets" who he is...no judgement, no blame. Some children need this extra "being held" by the teacher, in the earliest years. They feel adrift, and the attention and connection can help to anchor them. I hope you find a good teacher next time.
The best actions you can take with your son is to assure him that her is loved and that you are there to help him change his behavior (not him). A child this age may not make a distinction between his self and his behavior, so making it clear that he is good, and it is just his behavior that needs to change (and that changing his behavior will make HIS life better) needs to be stated over and over.
Your son is fortunate to have parents who seek out help for him. It must be confusing and worrisome for you right now, so I hope you get some answers quickly. It may turn out that there is an easy solution to this situation. I hope so!