"How does the cost of living in Scotland compare with the US?"
One thing is for sure, living in Scotland will be a unique experience which means that there are many different factors that can be taken into account to compare cost of living in one country versus another and many can be related to your own preferences and requirements.
In general the cost of living in Scotland will vary between town and country, with the cities being more costly. Items such as petrol (gasoline) are more expensive here than in the US, whilst other things will cost less. That said, the most comprehensive study that I've found is "Prices and Earnings Around the Globe". The only UK city it includes is London and comparing the US cities to London and ranked, with the most expensive first, the list is New York, Chicago, London, Los Angeles, Houston and Dublin.
The average for Scottish cities will be lower than the cost for London and comparable to Dublin. To give you an idea of the relative cost of living in Scotland compared to London, I've looked at apartment rentals. In regional trends across the United Kingdom the reported average weekly rent in £(sterling) for a private sector apartment was £141 in London versus £71 in Scotland. The average selling price for a home in Scotland versus London was £67,731 compared to £168,783 respectively.