Aside from Edit and Delete, I cannot find any way to add other "managing" commands for the users. 

Is it possible for them to Move an image from one category to another? If so, where do I set that?

Are there other commands than Edit and Delete that would make sense to allow users to do? If so, which ones and how?

 

Is it possible for them to Move an image from one category to another? If so, where do I set that?

No. Move is a moderator permission. You can certainly achieve this with a custom plugin. 

OK Thanks. It was not a necessity. If needed, they can ask an admin or moderator. I just wanted to make sure I was not missing something.

@Adriano FariaI'm a superadmin and I can't see where I can move an image from one category to another.  

 

@Adriano FariaI'm a superadmin and I can't see where I can move an image from one category to another.  

There's no "superadmin" on IPS4. Admin is a member who can perform actions in ACP. Moderator is a member who can perform action in front end. A user can be a moderator and not be an admin as the user can also be an admin and not be an moderator. It depends on how you set permissions... That's why you have a ADMINISTRATORS and MODERATORS separated page on ACP.

Again, you need to be a MODERATOR, and not ADMIN, to move things on front end.

Got it. I was also a moderator but I had to change the restricted moderator settings on my profile. 

Thank you for the clarification and education!

Thanks @Adriano Faria for assisting with these issues. Much appreciated.