close confuses aliases, "open with..."
 

My clone (on a USB external drive) causes my "open with..." menu to get confused and show duplicate app entries, and I have had aliases on my dock get confused about where they should point. I assume this is because the cloned apps are all visible on the mounted drive.

How do I stop this? Should I have the drive unmounted unless it is cloning? How do I automate this? Is there another solution?

Try rebuilding the "Launch Services" database with Onyx or Cocktail. This is something that Launch Services does on its own -- it searches for programs on all attached drives, and you get multiple entries.

You can try ejecting the drive (and you can leave it ejected if you'd like, schedules will mount/copy/eject automatically).