I'd like to know whether a 4,1/5,1 Mac Pro with dual processor board can actually work with just one CPU installed.
It will work, but it will make the CPU fan on the installed processor run at full speed. When I upgraded the processors in my '09 I made sure I had one running correctly before adding the second and discovered the issue. Fan noise is quite loud. You might be able to limit it with a third party fan control app, like 'Macs Fan Control', but I'm not positive and am unwilling to test it by removing one of my processors. :)
Also, you will be limited to the four RAM slots of the installed processor.