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?

1 minute ago, Carole Asselin said:

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.  

Just now, TracyIsland said:

@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.