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)

This is a procedural shader, made with and for Blender, so it will only be 'editable' in Blender. This means that you will only be able to modify and play with the settings in Blender.

Once satisfied with the cardboard look on your model, you can bake the texture maps on your UV unwrapped model and then take it to Unity, Unreal or any other game development suite of your choice that supports 3D. But you won't be able to change or edit the looks once in Unity/ Unreal.

Yes, Cardboard is compatible with both Cycles and Eevee rendering engines.

Yes, mostly. Everything except the recycled paper texture and scratches is procedural.

Yes, with the studio lighting setup. :)

No, the shader uses 'object' texture coordinates, thus, you need not worry about UVs.
However, if you wish to use the UV coordinates instead (for a UV unwrapped model), simply turn the last slider on each Cardboard node from 1 to 0.

You need to apply scale.
Select model. Hit 'Ctrl+A'. Select 'Scale'.