When a Category is first created, it has a blank Icon and Logo Image. You can update the image from the Details > Edit view. When you download a new image file, it may not show straight away in the Image display. This is due to browser caching of any older image. If you got a success message, the image has been updated and will show after the cache has been refreshed.
If you delete a Location which is the last Location to use a particular Category, that Category will not be deleted, but it can only be viewed by the Category Owner from the User Content view. This is intentional to allow Category reuse, and because the owner of the Category may not be the Location owner, and therefore may not have the authority to delete the Category.