If your going to run it, run it from the beginning, at least in IMO, otherwise you risk a "large" chunk of something that falls off from the cleaning choking the pump. (I always run mine)
I rinse after I clean, but nothing excessive. (people does vinegar... may mess with your params slightly but I've never noticed anything)
I've noticed that even if I unplug my skimmer overnight I get a new break in period.