How to get image name, mime type, width, height, file size from post custom field?

You can get the file from your Url like this:

//get the URL from your Metafield, $post is your current Post
$image = get_post_meta( $post->ID, '_your_image_url_meta', true );

//set up a query to search your posts table for the Url         
global $wpdb;
$query = "SELECT ID FROM $wpdb->posts WHERE guid = %s LIMIT 1";

//prepare and launch the query
$result = $wpdb->get_results( $wpdb->prepare( $query, $image ) );

//if there are results
if ( count ( $result ) > 0 ) {

    $image_id = $result['0']->ID;

    //get the image data
    $image_data = get_post_meta( $image_id, '_wp_attachment_metadata', true );

}   

All the information you need should now be available in $image_data. Keep in mind that you may need to change the script a little bit – for example if your Authors put in a thumbnail-url, you have to clean it first and change the query to a LIKE statement.