When I click on the tab "Categories" when editting a product, nothing shows up in the main content area(see screenshot).
This might be a problem with the categories specifically however, because we are also experiencing problems on the Manage Categories part of the site as well. With this particular issue, we see the editor tools appear for a second then they disappear and can not be found in the <div> they should be inside of.(see next screenshot).
Again, the form shows up and then disappears. I have seen this phenomon in some other posts elsewhere but have not yet found a solution.
In that case; could you disable all your extensions (on a development site) and try again and see if the issue persists?
If it doesn't - then enable them one by one re-testing after each step until you find your problem extension.