I believe that part is a known bug (dragging and dropping into the post while using Firefox).
OK. Now that I know it’s a FF problem, I found the previous thread on it as well as the ‘wontfix’ status on
http://trac.wordpress.org/ticket/2612
*sigh*. I’ll mark this as resolved, thanks for your help!
OK, after hours of frustration and delving into the code I finally got this to work after hardcoding the paths in wp-includes/functions-post.php -> wp_upload_dir().
After some more testing, I realized that this change didn’t actually affect anything; uploading and linking work when the links are created from the pop-up option menu for the attachment. But if I create them by dragging & dropping in Firefox, the ‘/content/’ part disappears again :-/ Now where the heck does THAT come from? I couldn’t find any incorrect references on the page or the generated JavaScript.