It's fine to use the URL to get to places you can't get to with Moodle's interface but what worries me more is that you are saying students and guests are able to edit from there too -that should not happen under any circumstances. Are you sure you tested that out? Also you say you edited so many permissions - that might be a cause of your problem -You as admin and a course creator of that category should be able to see the edit button but regular teachers wouldn't normally have category page editing rights anyway. How about setting the permissions back to what they were (leave admin alone completely) but course creator, teacher and student permissions. Nine times out of ten you don't need to edit the defaults anyway. The other possibilty is if you have a customised theme - try switching back to the standard theme and seeing it that makes a difference. If none of those work and you want to DM me a login I'd be happy to take a private look. (I have a distant memory I might have been to your school last year?)