Is it bad to refrigerate cut up onions?
Restaurants store cut onions refrigerated all the time. They will try to use them in one shift but they can last longer if needed.

They will go soft after time and lose flavor and crispness.  As far as going bad it would take over a week.  It will be too soft before it actually goes bad.

At home I try and cut on demand. I will halve and make the vertical and horizontal cuts but only finish the dice on demand. My experience is vertical and horizontal cuts on demand causes more damage. Most of the damage is on the exposed surface, which stays the same either way. This assumes proper technique, good knife skills, and a sharp knife.

If you want slices then easy.  Just take a slice on demand.  

Store them cut side down in a sealed container.