Love you picture, an airplane at last!
Somewhere inside the FMC is a prompt to display IR positions on-screen (green rings). Use it to investigate and see what is going on the next time you are returning inside NAVAID coverage. [I am definitely not saying to switch-off RADNAV update to test this, and never done it myself
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Sure, the IRs are stand-alone and receive no outside inputs. Other than that you're asking the right questions. Check
The Boeing 737 Technical Site - Home Page which is a pillar of wisdom that started well in the pre-NG days (the book is a must have).