No. While you will want a relatively warm coat with you for evening, an "American" winter coat would be overkill.
Yes! Notwithstanding my criticisms of athleisure in Europe, you can wear sneakers in Scotland.
You probably don't need them! Rain chances per day are relatively low, and when it does rain, it's just just a few drizzles or a bit of rain.
Thankfully you shouldn't have to worry too much about rain during your September trip to Scotland. That said, I would pack a small, cheap, foldable umbrella for days when you need it.
For under that jacket for daily wear, you're going to need some good base layers. I'd generally wear long-sleeved shirts and light sweaters.
Yes! You are welcome to wear jeans for your day-to-day tourism in Scotland. Make sure your jeans are in good shape, without holes or too distressed.