Hi!
What you’re trying to do actually already exists in WP. Since you’ve created a custom post type with a custom taxonomy there’s an archive (provided that you have set archive to true when creating the post type). So you should be able to reach each taxonomys posts by having an url something like this:
http://www.myurl.com/q_and_a_category/design
Wordpress will use archive.php per default to display these but if you require customization and dont want to affect the regular archives you can create a new template:
taxonomy-q_and_a_category.php
If you want specific templates for the terms you can do:
taxonomy-q_and_a_category-design.php
etc.
If you still want to do it in the way you’ve done so far (i would not recommend it) you need to replace the meta-query with a tax-query since the taxonomys terms are not meta values..
Read more here: http://codex.wordpress.org/Class_Reference/WP_Query#Taxonomy_Parameters
