This is a really tricky predicament. On one hand your son sounds like a typical 7 year old (I worked in a classroom with 30 of them and saw all the behaviours you described on a daily basis). But on the other hand, you would like to think that if he does have ADD you could set up support for him as quickly as possible.
I will say that developmentally, seven is an age where children become more self-aware, compare themselves to others, care about fitting into and forming groups, and judge their peers as well as being judged by their peers. So his anxiety could be a part of normal development. Only you will know if he is constantly anxious or whether it is something that has developed more recently.
As for his friends – God help the person that comes between two quarrelling sisters! Perhaps try to encourage 1-to-1 friendships for a while rather than groups of friends, so things are not so overwhelming for him.
Try to find out what it is he thinks he is failing at? As an academically gifted child, could it be that he has been teased about this at school? Perhaps he is being held to a higher standard than others, because he is so bright? He may be advanced cognitively, but please consider whether his social, emotional, physical and language skills are developing well too? These are just a few of the many things I would try to eliminate before I concluded that he had ADD. Surely his school will have observed him and will be aware if something seems to be holding him back, so I would book an appointment to talk to his teacher about your concerns.
Perhaps we ask too much of our little ones? In the grand scheme of things they are really still babies at 7.
Please don’t take offence to what I have said or think that I am trying to dismiss your concerns. You are clearly an attentive parent with your child’s wellbeing at heart. You know your child best and if you have a gut feeling that something isn’t right, then you should probably act on it. Hope this helps. x