Do you have any suggestions about equipment and/or techniques?

If your goal is to reveal damaged areas or truly demonstrate that a wall isn't damaged, aim your light so that it shines along the wall instead of straight at it, and aim the camera straight at the wall. In strong directional light coming shining along the wall, any holes or depressions in the wall will show up dark because the light won't fill them in, and anything proud of the wall (like nails or damaged areas) will cast long, noticeable shadows. Don't use a light mounted on the camera for this as it'll fill in the shadows and dark spots that indicate damage.

On the other hand, if you want to make the walls look as flat and damage free as possible, soft light shining toward the wall in the same direction that the camera is pointed will help fill in the shadows. You can buy a soft box or large umbrella to diffuse the light so that it's less directional.

Soft, even light probably makes sense for drawer interiors, where you probably just want to be able to see into all the corners and confirm that the drawers are empty and clean. On camera flash with a good flash diffuser or small soft box should work well here.

I would like to shoot at as lower ISO as possible to reduce granularity when the images are blown up on a screen.

The 24Mpx images from a D5200 have much greater resolution than any computer screen, so you should be able to zoom in quite a bit before any noise shows up. Also, you're probably not going to much noise at all shooting at ISO 250, so I don't think it probably makes sense to buy a faster lens just to shoot at slightly lower ISO.