I took a long look around, and didn’t find a good concise explanation, so…
How does one link one Object’s Bevel Profile properties with another’s, so that altering one alters them both? Dynamically, interactively, “live”? Is one the Master and the other the Slave, or is it transitive?
If you mean the traditional “Superellipse” profile, then either with a driver, or via alt-dragging/alt-clicking (assuming everything you want to modify is selected and has the Bevel modifier with the same name).
If you mean the custom profile curve, then… you don’t. It doesn’t even have Copy To Selected (and that is not a bug 
), even though you can ctrl+c/ctrl+v it 
I tried, but they didn’t listen.
…To the other question, I’m reasonably sure they’re properties.
Huh. I thought there was a mechanism in place to dynamically link all similar properties to other, similar items (meshes, curves). I’m sadly surprised this isn’t the case.