He sounds a whole lot like my younger daughter. I doubt that she has ADHD (which I think has become a catch-all for any type of problematic behavior and is WAY over-diagnosed), but she IS gifted academically. Her little brain just goes and goes and goes, 24/7 without pause. This can lead to LOTS of physical activity, LOTS of talking, LOTS of being separated from her table at school. However, her academic work is near-perfect. We moved her from Pre-K to Kindergarten in the beginning of her Pre-K year because the Pre-K teacher saw that there wasn't much that could be done to challenge her academically. She flew through Kindergarten and 1st Grade, and is now in 2nd. She has either 99% or 100% in every subject. Super smart, but with that comes thoughts and feelings that she is not always prepared to deal with. She tends to be very emotional.
What has worked for us is to be very consistent and no-nonsense with discipline. If she is being obnoxious, we tell her so, and she adjusts her behavior. (However, if she's really wound up, it can take a few reminders to reel her in.) If she melts down, she knows she has to go to her room and sit until she can calm down. She does best when she is highly challenged and has complicated things to think about and do. When she's bored, we run into trouble, so I work to make sure she has lots to do. Her teacher is pretty good about this as well.
I guess for me, I figure, some kids are more of a challenge than others. If my child's challenge is being a genius, then so be it. I'm definitely not going to medicate the smarts out of her because her personality can be inconvenient at times. She is who she is, and she's going to be this way her whole life - so she needs to know how to manage herself and respond appropriately to different situations, thoughts, and feelings. So, that's what we have been focusing on these past few years. I'm not sure if this is the answer in your situation or not, but just know you're not alone in this.