He sounds a little bit like my DD5 was at that age, but I also notice some differences, so I don't know if this will help or not. She was/is very sensitive to new situations and new people. She would not talk to other adults and very little to other kids excepting her close friends. She is also extremely sensitive to loudness and chaos.
Anyway, we realized early on that academics wouldn't be a problem at school, but the social aspect would if we didn't do something to improve the situation. I enrolled her in lots of classes. She was my first, so this was easier than it would be for subsequent children. I started with short classes in which the parent would also participate, like parent-tot gymnastics and parent-tot story times. After we achieved some (small) degree of comfort with that, we moved up to short classes apart (while keeping the parent-tot classes for continuity). We had a 2 year old toddler 'preschool' type class at our city rec center, and it was 45 minutes long. She was almost 3 when we started it, and they let us stay in it until she was almost 3.5 since they could see she wasn't ready to move up. Let me tell you that this was painful, she cried for most of the first month. But it was 45 minutes, it wasn't the end of the world you know? It did feel like it.
Anyway after she was reasonably comfortable there (barely participating, but not crying) we started 3 year old montessori preschool 2 half days a week. This was a big turning point for her. It took a long time, but she really did well there and in her 2nd year there just blossomed. The montessori school we choose was very low chaos even though there were a bunch of students in the class. It was so peaceful and orderly and she could limit her interactions if she felt she needed to. The mixed ages worked in our favor too, since an older girl took her under her wing. During this time we moved her up from parent-tot gymnastics to regular preschool gymnatics without parents. She kept the same (wonderful) teacher the whole time, and I think it was a major contributing factor to her improvement. We did art, storytimes, classes at the museum, gymnastics, sports, and swimming in addition to the preschool and toddler preschool. Maybe some others in there that I have forgotten.
It took a total of 3 years and a lot of work, but she is now ready for Kindy. She is still shy, but will talk to teachers and will talk to the other kids and will run off to play in slightly chaotic environments. She still avoids moderate to very chaotic situations, but honestly I'm not comfortable in them myself, so I don't blame her. She is still very, very sensitive, but many of the problem situations we encountered when she was younger, she has learned to deal with much better and can 'keep it together'.
My DD had a stammer like you describe at that age as well. It was as if she couldn't get her words out fast enough and her brain was 20 miles ahead. She has mostly grown out of it, but does still stammer when she is extremely excited or very nervous.
I did not ignore my DD when she would cry because I could see she was genuinely distressed. Obviously I was not there when she was in classes though.
Ironically it is now the academics I am more worried about than the social aspects (that she won't be challenged, etc).
I hope something in that huge novel I wrote helps...