My son was, and in a lot of ways still is, exactly like your DD at that age. In fact, we also had the OT eval done when he was four, and they determined that he had sensory processing disorder. His pediatrician's exact quote was, "Of course, he has sensory processing difficulties. That's what makes him a four year old." She said that it's a normal part of development, and if you take your child to be evaluated for SPD, he's going to have it! Like the car thing.
That said, I think there are many times when therapy is warranted for SPD, but I'm not sure it makes a huge difference for cases like yours (mine). If our children received the therapy daily, it would help for a couple of hours (sometimes a little more) after the therapy session. But, going once or twice a week only seemed to help our ds for a couple of hours on those particular days. We had to do the stragegies at home too to see any results, and we're still up in the air on whether or not it helped.
I see it as beneficial for severe cases, not on-the-fence type cases like yours and mine. It could be helpful if you wanted to go to therapy for a few visits with you daughter so they can teach you some calming techniques to try at home. I think you can find them online or in books also. If you can get her to a playground to climb and swing, that would be great. Also, any heavy muscle work like carrying a laundry basket, carrying books to the bookshelf, etc are great. Wheelbarrow walking, always fun, is a great exercise that taxes the muscles. When the muscles are worked, it helps her relax. It does that for all of us.
I wouldn't search out a formal diagnosis for this (just my humble opinion). What you are seeing sounds very much like overexcitabilities and appears to be common for gifted children.
Our ds6 grew out of a lot of those behaviors, but he still has some of them (pushing on my legs or against the table/wall, climbing on the sofa, making loud noises, spinning sometimes). He does much better, and the behaviors are almost non-existent, if he's had lots of physical exercise.
Personally I wonder if we see a lot of this now because our kids aren't able to get as much physical activity as we did as children. We used to ride bikes for 6 hours straight or roam our very large neighborhood on foot. We played kickball with the neighborhood kids. DS isn't even allowed to play in the front yard! I also wonder about the amount of TV, computer, video game time they have vs. what we had. It's tough to avoid as it's such a pervasive part of our culture now. We're very guilty of this at our house, although we do try to limit daily electronics time.
I'd definitely recommend using the sensory techniques at home. They are good for all children, regardless of whether or not they have SPD, overexcitabilities, etc.
We are now getting ds tested by a developmental pediatrician for possible ADHD. We've always attributed it to overexcitabilities, lack of challenge, possible sensory issues, etc, but it's beginning to look more and more like ADHD + overexcitabilities in our case.
Try not to worry; it seems like what you are seeing is very typical with gifted children, at least according to these message boards. Many days, I think it would be easier if ds was just a regular Joe with average intelligence, but I try to see it as a gift that comes with a price.
