Please don't feel like a failure. You are obviously a very caring, loving mum, which is why you are here asking for help.
Obviously none of us can say whether or not your ds is on the spectrum or perhaps has dyspraxia or a communication disorder, but I can tell you he sounds a lot like my ds at that age. He is nearly 10 now and was diagnosed with Aspergers in January last year and then tested on the cut-off point for dyspraxia a month later.
When he was five he pretty much played alone in the playground and was horribly bullied. Things improved a bit in year 1 and by year 2 he had a best friend, who remains steadfastly loyal to him to this day. 
He also struggled a with sitting on the carpet and had to lean on a table leg for support.
He was slow to start to read at that age, but by the end of year 2 he was made the youngest free reader the school ever had and was given a reading age of over 12 iirc, could have been 13 - well as high as the scale they had could test him anyway. (This has since settled and he is now high-average in terms of reading skills.) Whereas he has always struggled with writing and fine motor stuff.
Re your point about him being fairly normal at home. There is an expert on Aspergers called Tony Attwood, who once said that as soon as you put someone who has AS in a room on their own, their AS disappears and the same is often true within the family for children who are higher functioning, because their family understand them and are used to who they are and how they feel and behave in different situations. We tend to compensate without realise we are doing it and to not notice things about our dcs that other people might find odd.
When our dcs are at home with us, they are at their most secure and relaxed and therefore less likely to exhibit so many of their coping strategies (hand flapping etc) a lot of the time and will of course be more relaxed in their social interactions. My ds can give eye contact to the close family - although he doesn't always, but not to people outside of his close circle.
It sounds like the school is proactive and the teacher supportive, which is great. As others have said, she needs to take the lead in quashing any name calling and bullying, especially as they are such young children. I would have thought perhaps buddying him up with a slightly older child at playtime might also be a possible way forwards for him, as the little ones at school always think its really cool if someone has a bigger boy for a friend.
In our case, we went to see the GP without ds first and took a list of our concerns with us to discuss. Then we booked another appointment and took him back to meet her, at which point we were referred to the Paediatrician, who then referred us on to the multi-disciplinary ASD team who diagnosed him.
Good luck and let us know how it goes.