When using it for my locker, I have the cover on. When airing up I pull the cover for two reasons... First so it can breath/stay cool, and second because my air chuck wont allow the fitting to work with the lid on.
I mounted the single with just two L Brackets from Lowe's. I only use mine for the lockers, but I leave the lid closed. I left the carpet in to deaden the sound a little and it isn't bad at all. If you are going to do the twin and use it for an extended period of time you would want some air moving around it, lid open, to keep it cool.