My son does have ADHD, and he does seem similar to your son. So, it's possible. If you have concerns, talk to your doctor about them and do ask for a referral to a psychologist or psychiatrist who specializes in children and behavior disorders. Whether you tell the docotr that you think your son has ADHD or not isn't going to determine the diagnosis. But it might give the doctor a place to start.
Right now the best thing you can do for your son is to keep a notebook. (I have one for our son, and I bring it to all parent meetings.) Write down things he does that concern you or give you pause. Doesn't matter if you think they are related or not. Just make a list and add to it as things come up. Roaming in the classroom, not wanting to sleep at night, foods he likes, foods he doesn't like, habits, etc. All of these things will give the doctor more information to help identify your son and his needs.
Sleep is a big deal. A very big deal. If your son has trouble settling down at the end of the day, he's not alone. But he does need his sleep. You might want to ask your doctor about giving him Melatonin. Our son's psychologist recommended we do this. He said sleep is super important, and if our son is having trouble settling down at night, Melatonin might be a huge help to him. It's not a drug, and it's not going to help him stay asleep (or cause him to have trouble waking up like sleeping pills can). It just helps him to relax and actually get to sleep.
Remember, of course this is the way God made him! If he were short, that would be the way God made him. If he had juvenile diabetes, it would be the way God made him. Just because this is the way God made him doesn't mean that there aren't ways to take this amazing kid that God made and help him to function better, learn better, socialize better and in general have a great life!
I often hear people say that he's too young to be evaluated for ADHD. This is simply not true! He is not at all too young to be evaluated. A doctor might say he/she is not ready to diagnose him with ADHD (or whatever fits him), but an evaluation can serve as a baseline. Our psychologist didn't diagnose our son for 2 1/2 years. He suspected ADHD, but he wanted to observe our son longer and listen to what his teachers said over time. He took that diagnosis very seriously!
It cannot hurt to have your son evaluated. The doctor might give him a diagnosis. The doctor might say he does not believe there is a diagnosis and that your son is simply a young boy who needs time to mature. You might (and most likely) hear something in between - this or that is suspected and should be observed for x amount of time. Either way, wouldn't you rather know?