The Shwifty Shaders are supported by and will work with both rendering engines, Eevee and Cycles. Feel free to work in any of them.

No, the shaders use 'object' texture coordinates, thus, you need not worry about UV unwrapping at all. :)

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

These shaders are made with and for Blender, so they 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 shader's 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 texture properties once in Unity/ Unreal.