tape residue is better removed, at least in my opinion, with mineral spirits, just wipe it down.
I use a corded drill, mounted on a board for the sanding mop, the mop hangs over the work bench. I did this mostly cause I do not believe the bearings in a drill press are suited to lateral pressure, and it allows me to leave the mop mounted. Of course the corded drill bearings most likely have the same issue, but with corded drills less than $20 its a cheap replacement not to mention easier. In my set up, I found that if I throw a sheet down under the mop, I no longer lose a piece since that makes it easier to find, corollary to Murphy's law.
I actually discovered that trick working on harleys, there are some tiny parts is some areas, invariably I would drop one into never never land, started putting down a sheet, and no problems since.