Not a frequent user for Photoshop, there's always this shadow when I map my art on a sphere. Is it possible to remove it? And is it possible to keep my art there while checking it out on the sphere?

This looks a lot like you have the lighting set to hard lights without the middle infinite light turned on.
Best way to fix this would be to go to Window>3D; Click on the lightbulb on the top right in the new 3D window and set the Preset to White Lights and make sure all 3 Infinite Lights are enabled. Light Type Infinite too.

Hope that helps.

Thanks Jebus. Instead of changing preset to white lights, I completely changed the default colour of grey to white and that seemed to solve the problem. Thanks.

Do you know the answer to this: And is it possible to keep my art there while checking it out on the sphere?

Not sure if I completely understand, but if you look in your Layers panel and find the one for the sphere you can double click on the name under Diffuse. That'll open up the flat for you to edit in another window. It will update the sphere with your edits and vice versa.