Mid September is the best time to visit Scotland - once the festivals are over, it's usually very quiet and relaxed. Some restaurants may be closed for holiday, but everything else will be open - and much less crowded. The last day of the International Festival is August 31 this, so any time after that is fine.
Weather is a toss-up. It can be sunny and 70s or raining and 50s. Up in the Highlands & Islands you can have 40s (or occasionally worse) at night by mid or late September, but Edinburgh tends to be mild. Pack a lightweight rainjacket and layers.
We don't really get a spectacular fall like many parts of the US. Some years it seems like the trees have all their leaves one day, and then the first big winter storm comes in and the next morning all the leaves are on the ground. Early September will probably still be summer green, but you never know...