i take the lid off when I listen.  Have heard that is recommended, to not have the big resonant surface area of the dust cover on the turntable while playing.  Makes sense.   though I can't really say I can hear a diffference.  If I do, it's a "sighted' comparison so I'm not sure I actually hear the diff.

I remove the lid on my TTs when playing vinyl. IMO and the opinion of my ears the sound is better without the lids. Leaving the lid on and up or down means you have a large resonant plastic tub attached to your TT.

Yes,The main concern is that the sound waves from the speakers would vibrate the dustcover and transfer vibrations to the arm/cartridge/stylus. Everything vibrates to some extent including the turntable itself and the surface it sits on, but the dustcover being large and light would be more susceptible.

would be interesting to compare listening to records with headphones only (no speakers)  dust cover on/off. That would eliminate a big variable, and could see if cover affects sound based on much smaller vibrations coming from turntable itself. 

The lid, when down, picks up vibrations from the air and transmits them to the turntable at the contact points.

With the lid up, this does not occur nearly as much.

The best thing, however, is to remove the cover. My old Harman-Kardon turntable was set up so you could just slide it out of the hinges and set it aside.

You can also just eliminate the hinges and set it aside when using the turntable.