<What is the function of the gray valve at the top of the rear heater core?
Is this an emergency shut off? Or some kind of temperature adjustment?>

That valve at the top of the rear heater core is a bleeder valve.  I guess
so you don't have to bleed the main coolant system.  Useful if you have to
replace your core like I had to about 6 months ago.  It's actually quite
easy.  Just clamp the inlet and outlet hoses and insert the new heater core.
Then I unclamped the lower hose, switched the lever to turn on the heater
core and started the engine.  Air came out, then coolant and bubbles and
then just coolant.  I then just closed off the valve.  Someone suggested
driving up and down a bunch of steep hills to let the self-bleeding system
work (wherever that hides out!) and it all seemed to go OK.  Anyway it
worked for me.

Good luck

Rich  90 Westy