Dust cover up or down?

Off. It literally opens up the sound when it's open or removed. If you don't notice much of a difference, you need better gear.

Bottom line: If you don't notice much of a difference

I don't use it at all. In fact my Avid Ingenium does not even come with one. On my Music Hall MMF-7.3, I even removed the dust cover hinges....

People spend tons of money to reduce any adverse effects on their turntable caused by vibrations. Removing the dust cover while playing a record is probably the cheapest way to reduce microphonic vibrations that will cause distortion in sound. Some cheap turntables with cheap cartridges may not make much difference with or without a dustcover because the sound is already pretty bad, but very revealing high quality cartridges may suffer from such vibration especially when you listen to music loud with the dustcover up. The dust cover will pick up the sound wave, however small, which may well cause vibration.

There are no absolutes. It totally depends on the turntable in use. Use your ears.

Don’t have, and don’t use a dust cover. 
 

Will likely get one made for my TT to keep the dust off when not in use, but seeing as how it is used daily, it will be spending the bulk of its time stored near by. 
 

Suspect if I get  one made, it will be used if I go away for an extended time.

A friend of mine leaves his on all the time, which is easy as it is hinged. Opens it, puts a record on, closes it. 
 

Seems to me, as the price of a TT goes up, the presence of a cover becomes a rare thing indeed. Especially one that is hinged. Can’t think of a single ultra high end TT that has a hinged cover? Would love to know if there are any?