It sounds like your issues are down to permissions. Especially given you state that there is a problem with gallery items being accessed, and it saying they are not present. Are you using a different instance of PHP? By this I mean suPHP for example, where you wasnt before. I ask as the permissions would be different in that case.
In answer to your questions
You 'may' be able to use the new, but I would advise on using the original. Especially since you already have issues.
You can clear the contents of that folder, with the exception of the index file. If its any problem there, then that would resolve it
While you should be using INNODB tables for all tables, its unlikley that would cause the issues you are describing.
Make sure of course any constants.php file and htaccess has no references to old locations. The constants.php file especially if you have one