Drywall – use concrete screws to mount storage racks through drywall into wall studs

Looking at your wall photos, can I assume that seeing hinges on both left and right, the doors (2) meet in the middle? first we have to make some assumptions based on the yet unknown. if the walls are always dry, seasoned and pose no moisture or condensation issues, there is a couple of ways you can go ahead and finish he walls with drywall. i would consider putting a 1X6 up to a 1X10 finish pine board in the corners. This should clear the hinges. You can then install 1X3 strapping on the concrete wall with PL400 adhesive and short ramset nails or concrete screws. You can then decide if you want to install any blue moisture proof rigid insulation between the strappings. Then install a vapor barrier such as 4mil plastic over the strapping and then install drywall with 1 or 1 1/4 inch screws. You then can finish your drywall and build a neat trim at the ends onto the finish pine boards. The other alternative is to build a regular frame wall out of 2X3's or 2X4's and mount to floor and upper joists. This will allow more insulation and still allow doors to open almost fully, depending on the width of the pine trim at the corners. The wider the trim in the corners, the more fully the doors will open without being restricted by the new wall. If moisture on the walls is possible, you should really consider using Dry-loc or some suitable waterproofing before you even think about closing the concrete in behind a wall. Mold and mildew with ruin your good looking wall and potentially cause foul odors and a possible health hazard.

You actually have a couple of options in this case. The easiest option would be to simply use 12 ft sheets of 5/8 fire rated sheetrock and rent a lift to do the ceiling with. A lift is very easy to use and almost makes it a one person job. Personnally, I'd also use 5/8 on the walls as well. As BMitch already mentioned, mount the wall sheets horizonally and stagger the vertical joints. You can use 2X4 nailers, (fire breaks) at the 4ft OC point, but not really necessary. You will not see any bowing or cupping with 5/8" at 24" OC on walls.

The second, and BEST option would be to install 3/4" strapping 12" OC across the field of the ceiling, and around the parameter. This will give you a lot more surface area to screw the drywall to and make aligning joints a lot easier, especially if the 24" joists are not perfectly spaced or square. Putting up strapping is quick and easy if you have a nail gun. Use 4D ring nails. Ring nails will hold tight and not loosen over time.

Hint: Use a chalk line on rock to mark backer locations before you hang it, or you can also mark the location of your joists or strapping on the wall before you lift the rock into place. Secure the rock with a few screws to hold it in place, then use the marks to snap a quick chalk line marking the location of your strapping. This will assure you hit a solid spot with every screw. Ceiling screws should be no more than 16 inches apart.