When reading various threads, specifcially about noise and sensitivity with the HVX, I often come across people saying stuff like, "Well, these were run and gun shots, when the scene is lit correctly, it will be beautiful."

It makes sense in fully lit situations. I understand the concept of adding light where it's too dark and taking away light where it's too light to stay within the dynamic range of the camera. But what about low-key lighting? Does "correct" lighting apply there as well? What if I want the shadows to fall into black. Is it correct to leave the shadows out of the range of the senstitivty or should I be adding light to the shadows to bring it just within the dynamic range of the camera and then crush the blacks in post if I want it totally black?

Do I still follow the general rule of adding light to shadows and taking away light from highlights to have everything fit in the dynamic range when dealing with low-key noir style scenes? How does doing that affect noise performace? How would NOT doing that affect noise performace?

In the footage I've seen it seems I see noise in the barely lit portions of the frame, but not really any noise in the sections where there is no light (like the night sky). I just don't understand how "proper" or "correct" lighting will eliminate or diminish noise in noir type shots where the actor's face itself goes from highlight on one side to complete black shadow on the other. How can you control or elimnate noise in a shot like that?

I'm assuming you're talking John Alton contrast levels of lighting here. Although yes, you'll obviously never get the type of dynamic range that you would in film (at least for now). You can come acceptably close.
It really boils down to extracting the most out of the camera you can. With my XL2 I crush the blacks in camera (Sorry, I shoot everything 90% "in" camera) and set my gain at -3, shoot wide open and pump lots of light into the specific areas I want, molding the direction of the light with a the barndoors and or black wrap.
I avoid practically any noise doing this.
If the scene is very static, meaning no one is walking or moving quickly I also shoot at 1/24 shutter speed (gasp!).
Planning out the lighting and how it falls and on what helps as well. Well designed and creative lighting is important.
Just my 2 cents.