Tim, the only real difference between a crawl space and a basement is the height and a basement is heated.
Since you didn't mention that you had a concrete floor, this changes things a little. The first thing you want to do is check to see how much humidity/vapor is coming thru the floor. You do this by putting a piece of plastic on the floor and taping down the edges with waterproof tape. Come back in a couple of days and look to see how much water has condensed on the inside of the plastic. If there is any, you will have to stop it. Can't really advise you what to use for this as all my experience has been with dirt floors.
Yes, you can use this space for storage but remember that it will be subject to temperature variations and possibly still some humidity. I would be very careful what I put there if you have any question about humidity. Steel rusts, paper/books get mildew, canned goods can freeze.
Jay J, another regular poster here, dropped me a note about how much ventilation you should have in the winter. He feels, I think, that you should have a vent that you open in summer and close in winter. I somewhat agree but do not feel that you should totally seal it off. I think that you should have a small vent opening to get rid of humidity coming thru the walls or down from the floor above. A big question is pipes. Are there any pipes in the space?
I do not feel that drywall on the ceiling is a good idea. Drywall is dewatered gypsum on the inside with paper on the outside. The gypsum loves to absorb water and when it does, it expands and gets mushy. You want to let any humidity into the space from above and vent it out, while stopping any from below. You have to let any that wants to come down from above or you will end up with wet fiberglass. This removes all insulation value and is a great home for mildew. I would not put anything on the ceiling.
Hope that this answers some of your questions. Henry in MI