I'm thinking of getting a cheap printer/scanner combo: HP Deskjet 2010 and running the whole process in Linux. Are there any downsides to such a setup when comparing to more featureful scanners such as ScanSnap S1300 or the Doxie?
You will save time, headaches and money (in the long run, at least) by spending a little more up front by purchasing separate devices that are designed to do the job you want them to do.
I can't speak to the Linux aspect.