Yes, for Woolly to map properly on your 3D objects, they have to be UV unwrapped.
All the primitive shapes come pre-unwrapped with Blender now.
(The Earth models you see in the examples is a simple UV sphere with a texture from NASA)
Yes. Woolly is a fully procedural shader.
That means no image texture has been used for making it.
This gives you infinite resolution. :)
The Woolly shader is supported by and will work with both rendering engines, Eevee and Cycles. Feel free to work in any of them.
However, if you are noticing any artifacts with Eevee, then simply switch to Cycles for a final render (once you are happy with the node settings in Eevee)