Custom post types support comments by default. I’m not sure how you registered your post type since you didn’t include the full source but the codex has good examples on how to do that.
If the metabox is showing in your custom post type and you are trying to hide it you could click on Screen Options in the top right of your browser and uncheck comments discussion. This just hides the comment metabox in the edit screen.
Make sure you are including the comment template part in your single.php or page.php.
====== Edit 2 – Correct answer ======
After looking into this further it looks like something is overriding the comment status. Putting the below functions will do what you want
// Sets the comments to allowed by default
function turn_on_comments() {
update_option('default_comment_status', 'open');
}
add_action('update_option', 'turn_on_comments');
// Hides the metabox in the edit screen (replace post-type-here with your custom post type)
function remove_meta_boxes() {
remove_meta_box('commentstatusdiv', 'post-type-here', 'normal');
}
add_action('admin_menu', 'remove_meta_boxes');