I've made a jing to illustrate what I'm saying below.
It would seem that the Q&A plug-in WHILE POSTING the questions, is not communicating with the backend of WP and telling it those questions are available (or posted) for that category (I speak here of the categories created by your plug-in).
Basically, while questions ARE shown in the Questions tab, they are not being registered as such in the backend of Wordpress. Because of this, the (populated) categories are not showing in the sidebar on the front end (public side).
Site in question is..
NOTE: We have approximately 10 questions entered (in three categories) but if you click on the Q&A Plugin category "Penny Stock Basics" it shows only one question, PLUS the other categories are not being displayed at all.
SECOND NOTE: Do not pay attention to the normal WP category (unless it is a conflict?)
Questions and Answers
This will be deleted once the question/answers are ported over to your plugin.