Should I store root vegetables with or without the dirt?
Not really, the dirt's there because of:


Authenticity: dirty veggies make people think "garden-fresh"
Laziness: farmers don't want to spend much time cleaning their vegetables


Washing to remove dirt won't shorten the shelf life, but mechanically removing dirt (as in with a hard brush) might in some cases as it could remove or puncture skins or peels.