Eyeballs showing up on my sim

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The reason you're seeing eyes on the mesh is because of the way your mesh is UV mapped. 

Your mesh is not ready to import into Studio, anyway. Take a look at this page to see how TS4 hair should be constructed and study the mesh you cloned to see how it was put together. You should have three mesh groups. Delete every mesh group except the main mesh group—the one the game will display when your sim isn't wearing a hat—and UV map it. You must use this CAS map and place the UV maps in the hair section. Make the uv_1, vertex paint the mesh, and transfer weights. Then duplicate the mesh two times using shift + D. The beauty of duplicating a mesh is that the new mesh will automatically be UV unwrapped, weight painted, etc. Edit the meshes of the duplicates to make the hat chops. Assign cut numbers to match whatever you cloned. These are just basics. There is more you should do, obviously, and Mauvemorn is the one who can help the most.

Right now, you have six mesh groups. You need three. And the mesh group called "toonshading" has its normals flipped. It's inside-out. It's easier to see this kind of thing if you have backface culling enabled. Press N to open the side panel to enable backface culling (ignore the right side of the screenshot).



To fix flipped normals, select the mesh, edit mode, press A, W > flip normals. The "toonshading" mesh contains part of a face (as in eyes and nose, etc.) so I doubt this mesh should be in your blend file, anyway. Probably you should just delete it.