Can you paint multiple texture layers simultaneously? There are tons of textures on the web that provide different images for color/albedo, reflectivity, bump, normal maps, displacement maps, etc... Suitable for PBR shaders, for example.

I'd like to be able to paint on a mesh in 3D using stencil or clone such that all of the source layers get transferred to the appropriate destination layers subject to the UV map.  I.e. A single brush stroke transfers correctly transformed texture layers to my object.

Can Blender do this?  If not, are there any plugins for this?

I believe other programs (substance painter?) do this. But I'm hoping to stick with Blender for now?
I believe you are looking for a mix of Sculpt Mode and Texture Paint mode. They work similarly as ZBrush, allowing you to sculpt in high poly mode, export a normal map, and also paint the UV maps directly over the model. There is no need for extra plugins, Blender does that natively. Just click on "Object Mode" and try the other modes.

You might also try Sculptris, by Pixologic. It's freeware (gratis, but not libre), like a very bare-bones version of ZBrush.