It sounds like he does better one on one. Can you find a friend to come over to the house for pizza or a movie or whatever?
If he seems to have difficulties with social interaction, does he tend to take things literally as well? Is he quite a rigid thinker and finds it hard to be flexible. Do you feel that he isn't misbehaving or being naughty, but that he genuinely doesn't know what to do about social interaction?
If he has literal interpretation of language as well you can google Semantic Pragmatic Speech Disorder and see if that sounds anything like your son.
If it does then he may be mildly affected or have traits of autism.
Does he have any obsessions or are his range of interests limited?
Has he ever been seen by a Speech and Language Therapist? If not you could ask for a referal as these are the professionals who are supposed to look at social interaction skills. You could also ask the school to refer you to their Educational Psychologist for them to observe and/or assess your son in school.
I presume that he copes well with changes to routine and doesn't throw tantrums.
I also presume that he doesn't have any sensory issues either (eg. over or under sensitive to touch, smell, sight, sound, taste, balance or co-ordination.
I have a son who is diagnosied as high functioning autistic (he is 8) and he too makes comments that he is stupid, and that he always gets things wrong. He thinks he knows what to do or say but when he does it and it becomes obvious to him that it was the wrong thing he gets very upset and angry with himself. For someone on the autistic spectrum he has alot of self awareness. As your son is also very self aware and is making negative comments about himself this means that these difficulties are leading to low self esteem which is not good as it can go on to lead to depression.
I am not suggesting that your son is on the spectrum, but it is possible to have milder forms of it such as Semantic Pragmatic Speech Disorder, or to have traits of autism which may mean a child has difficulties in one area, but not enough difficulties in all the areas required to get a diagnosis.
Sometimes my son has hit other children, but what he was actually doing was he was pretending to be a jousting knight and he was running up to children with his arm outstretched and bumping into them. But, because he doesn't have good social interaction skills, he hadn't told them what he was doing or asked them if they wanted to play that game with him.
From your post it makes it sound like your son is trying to keep a child's attention or interaction just with him and if they try to get away from him then he hits them. Does he tend to try to take control of the game and always want it to be done his way? If so that could be seen as a rigid repetitive way of playing that would also tie in with being on the spectrum. Children who do this need the predictable outcome that they can only get if they are in control, and they may also have limited imagination and therefore may only want to re-enact things they have previously seen on TV or in films.
I don't know much about ADHD, but some children with that diagnosis can be OTT and get quite physical. Is he able to sit and concentrate in class. Does he talk all the time.
School are obviously concerned about him. What do they suggest? And as your son is also getting angry and upset with himself I think you should discuss these difficulties with your doctor and ask for him to be assessed.
If you can afford it there is also the possibility of paying for private Play Therapy sessions. If you find a play therapist who works with children with poor social skills this may help your son improve his skills as well as give you a professionals insight into what aspect of social interaction he is struggling with.
Does your son belong to any out of school clubs or childrens groups such as scouts? He may find it easier in a more structured type of environment.