I like your neurologist's approach. And yes, a lot of the symptoms certainly sound like ADHD. It will be worth your time to start reading up on ADHD to understand what may be going on with him. This medhelp link is a good starting point.
Also your post sounds like your son is now in second grade. He would have been 5 going on 6 in first and 4 going on 5 in K. And while the cut off age can be Dec. in some districts, many parents will wait for the following year. So some of this could also be maturity. Some could be frustration cause all the other kids are older and grasping things faster. If he does have ADHD, its the perfect storm and I certainly understand why he would be having problems.
I would follow through on the ADHD first. Also don't be afraid to look for a child Psychologist that specializes in ADHD. If your neurologist also has that specialty, that would probably be ok. But both of you will need counseling on dealing with his problems, and the child psyc might be better for that.