There should be no problem with the Button Bar icons as long as the 'jbcore' folder is intact and the 'juicebox.js' file is successfully loaded into the web page. The location of the files should not make a difference (as long as all the paths are correct).
There is clearly no problem with the Juicebox font files or server MIME types as the Button Bar icons displayed fine in your original configuration.
This leads me to think that the problem might be a caching issue (either browser or server side) as the problem seems to happen after you move a file and change a path.
I can't quite figure out exactly what's happening but try the following:
(1) After moving the file and changing the path, try completely clearing your browser's cache before reloading your web page to see if this helps. If browser caching is the cause of your problem, then it is very unlikely that visitors to your website will experience the problem as browsers tend to check for newer versions of files on each browser session.
(2) After moving the file and changing the path, try (temporarily) renaming your 'projext x' folder (this should not break the gallery as long as the path to the 'juicebox.js' file from within the 'index.html' page is relative) and then reload your web page. This will create a completely new unique path to your web page that neither your browser nor your server will have seen before so there should be no strange caching going on. If you find that server caching is the cause of your problem, then ask your web host if they could turn off server caching on your web hosting account.
I hope this helps and points you in the right direction.
If you continue to experience difficulties, please post the link to your web page so that I can see the problem for myself and hopefully help further.