Dick,
Several comments. It is NOT the DX vs FF that is causing this. I have several DX & FF cameras and lens (Nikon) and I have NEVER had an issue. Yes, there are metering differences, but the likelihood of that causing the problem you describe is very small. If you do a spot metering on one bright spot, you will get an incorrect setting, but this does not seem to account for your issue. your first picture does not have any uneven lighting.
Please check my initial suggestion. It seems to me that the rental lens has an issue with stopping down the diaphragm. Your latest comment seems to confirm this.
You said that the metadata shows aperture settings that are not in the lens range. This would seem to confirm that there are issues with the lens. either the electronics ( contacts or electronics), or the mechanical action of the aperture.
Again, try it in a full frame camera , if you can get access to one. do you have the same issue?
Just a general suggestion. I am not sure if you were using this settings, just to show issues with the lens, or if these are settings that you often use. but just in case you often use these settings, a few suggestions IMHO :-)
Stay away from small apertures, such as the f/36 of your test shot. you are generally much better off using an opening in the mid range of the lens, such as f/8 - f/11. Of course, this will vary depending upon lens. you can check specific lens tests for where your lens has the best quality.
But, you almost never want to be at f/36. ( depending upon lens & camera above ~ f/11 you may start to have issues with diffraction).
Also, the D7100 does really well at ISO's quite a bit higher that ISO 100. This means that (instead of 1/6 s, F/5.6,ISO 100) if you had shot the picture at 1/25, f/5.6, ISO 400, you would have a faster shutter and far less camera shake. at 1/6 you need a tripod. at 1/25th you may be able to get away with hand held. Or try ISO 800 and set the shutter to 1/50th. With the D7100, and depending upon what you are doing with the picture, you may not be able to tell the difference.
Likewise, for the 1/10 sec, f/36, ISO 800. You would be much better off with 1/200, f/8,ISO 800, or perhaps 1/100, f/8,iso 400
There are lots of discussions on this site regarding use of ISO, aperture & shutter speed.
A good link to check out equivalent settings is:
http://www.scantips.com/lights/exposurecalc.html