Yes - I have 82 inches of RO for door currently to the concrete slab. Of course I'll be putting in flooring - carpet. That will take up close to an inch. So I may need to shave a door just a bit. My question is - should I be installing the pre hung doors first - or can I sheet rock then put in the doors after?

Get some foam board or fabric in black & white & tack it up on ceiling to see what it looks like. You could also paint a test area. Dark colors usually close in a space visually. White matte visually usually lifts. Usually white matte is the way to go.

The basement ceiling height of the house we lived in when our kids were teenagers wasn't much higher than yours. My husband sprayed the underside of the main level floor matte black, covering all pipes and ducts. If you DIY, be sure to use a good respirator mask. Our son was the same height as your husband (now slightly taller) and my husband was just a tad shorter. Neither one was bothered by feeling as if they were about to bump their heads. We had no issues with dust. As for noise in the basement, I think it was only noticeable when someone overhead wore high heels. I didn't find that the sound of video games or movies in the basement carried upstairs any more than in a later house with a drywall ceiling. My family liked the look and I was happy to put the money that would have been spent on a finished ceiling toward other things in the house.

Exposed & painted joists & utilities is a fun look that helps ceiling height perception BUT a very difficult messy project, a lot of work to “tidy up” the wires, pipes, valves & ductwork into something presentable before spray painting it all. I wouldn’t go through all that in a house you intend to sell. Maybe the realtor can suggest this to potential buyers as a possible remedy they can do themselves and show them examples of other 7’ basement ceilings that have exposed joists. Pop out a couple of the drop ceiling panels in a good spot (somewhere without tangles of wires visible) and take a pic & measure the difference in height; paint a sample area of joists white, the realtor can show buyers that sample area and see if that might be an acceptable compromise.