By default the WordPress Media Library uses the same categories as WordPress does (such as in posts & pages). If you want to use separate categories for the WordPress Media Library you can use a custom taxonomy, this can be set under Settings → Media (or click on the settings quicklink on the WordPress plugins overview page).

Or if you have the plugin version 2.5 or older you can add this code to the file functions.php located in your theme or child-theme: /** * separate media categories from post categories * use a custom category called 'category_media' for the categories in the media library */ add_filter( 'wpmediacategory_taxonomy', function(){ return 'media_category'; } ); //requires PHP 5.3 or newer

Yes, you can delete the free version and then install the premium version. All assigned categories will be remembered.

By default WordPress assigns new posts and media to the category “Uncategorized”, this can be changed by manually selecting another category before you upload new media. Read more: How to set a default category in WordPress