Normal to people on here and normal to the general population may be very different. I am very sensitive to light, smell, noise, etc. and my DD actually seeks intense levels of noise and that is very challenging. Yes, I think there is the issue of excitabilities, but I think OT has done wonders in our home.
As I have said in the past, my DD4.5 is very spirited, intense, unable to stop talking / moving, reactive, etc. While her "issues" are different (she is the overwhelming force and noise, and doesn't retreat or avoid eye contact) she was very controlling, prone to outbursts, etc. alongside having very advanced abilities.
We took her to an OT and she has been going once a week for months for OT for "sensory seeking." If there is anyone there who can assess for sensory issues, it might help you to decide if sensory issues are involved.
My DD4.5 has actually been doing ten times better lately and I am not exactly sure why. We are going ahead with an assessment to rule out anything else (ADHD, Aspergers) even though those things are looking less and less likely lately since her behavior has improved greatly. She has found one good friend and that seems to have helped her. This friend is her age and does not seem to be gifted in any way, but she is a kind, verbal, imaginative child who is good with give and take so my DD4.5 is thrilled to have someone who wants to act out scenes with her since many of the three and four year olds she played with actually were still doing parallel play which seems absurd to me, but we had a hard time finding someone who would join and interact with her at the level she craved.
My friend's child sounds alot like yours and he also appears to be gifted,but not to the extent your child is. My opinion is this: no matter how brilliant a child is, he or she should learn to interact in a respectful way with people of different ages, abiliites, temperments and that brilliance does not excuse "manners." I know kids with Aspergers really struggle with social niceties, but even if my DD had Aspergers, I would work on teaching her appropriate social behaviors. That's just my two cents. I want my child to be a kind respectful person no matter how creative or smart she is. In fact we stayed in a pre-K with kids far beneathe her ability because they were socially mature, kind children and we are working on that social piece and although it wasn't ideal, and she is frustrated with the class, it was the best we could pull together for now.
I am not saying what is going on with your DD is abnormal, just that I think it is best to check things out if it is possible. I know you are in a remote area and that reduces the social and professional resources. Have you done much reading online on sensory issues?