The Auto Eye is composed by two meshes ( iris and sclera-cornea ), so to duplicate it correctly you need to select both meshes. The same applies to the Mirror Modifier, you need to add the modifier in both meshes.

To fix that just go to Eevee and enable the Screen Space Reflections and also the Refraction option.

Furthermore, I recommend open a new empty file, enable these options and save the Startup File to have these options enable by default in Blender.

Yes, you can bake Color, Specular, Roughness and Bump.

There are 3 tutorials covering this process to guide you.

Baking Maps
Setup Maps
Setup Materials