It might just be the late hours, but the register_taxonomy call looks wrong. Shouldn’t the second parameter be the post type you’re registering the taxonomy for? I.e. array('pdsh_posts').
And yes, you need to register each of the three taxonomies separately for the post type.
See the codex example, https://developer.wordpress.org/reference/functions/register_taxonomy/#comment-397
Edit
Actually, its probably not what I wrote above. Registering the cpt with the default category taxonomy, if that is what you want to use, should only need the following parameter for the cpt
'taxonomies' => array( 'category' ),
The reason why you’re getting not found on the category view is that by default the Loop only looks for posts, not your cpts’. You’ll need to add the cpt to the query with pre_get_posts
Example here, https://wordpress.stackexchange.com/a/114853/144392