The Fairlady Z left rear brake started making that "metal grinding noise" so I took the drums off both side. The shoes look great, all the hardware looks fine, no scoring on the drums ... any clue what might be making it off and on? Something gets in there perhaps, like a small rock or something?

Are the rear drums interchangeable or is there a "right side drum" and a "left side drum?"

You might have a bearing problem that is showing up with the change in loading from the brake engagement. The drums are the same for both sides.

Any way to quickly determine if it's a bearing problem?

You might have a bearing problem that is showing up with the change in loading from the brake engagement. The drums are the same for both sides.

Look for marks where the backing plate rides close to the drum. It's pretty easy to ding the bottom of the plate up to where it contacts the drum, or, as you suggested, there could be a piece of gravel stuck in there somewhere. You'll see light colored rub marks on the backing plate of the edge of the drum. If so, a pair of pliers or a small hammer will easily fix it.

Should the shaft have any side-to-side play in it at all when one grasps the hub and tries to move it up and down, or in and out?

Look for marks where the backing plate rides close to the drum. It's pretty easy to ding the bottom of the plate up to where it contacts the drum, or, as you suggested, there could be a piece of gravel stuck in there somewhere. You'll see light colored rub marks on the backing plate of the edge of the drum. If so, a pair of pliers or a small hammer will easily fix it.

Mine does a weird shaking when I brake coming from the rear. It didn't happen until I did my lowering springs and struts. I took a look at the drums and couldn't find anything

Took the rear brake drums off, sanded them, reversed sides, cleaned up the shoes, and all is well now. No rubbing sound. I think some of the brake grease may have overheated and dropped on the drum causing a discoloration.

Should the shaft have any side-to-side play in it at all when one grasps the hub and tries to move it up and down, or in and out?

No side-to-side and only .0057" end play.