I had this same problem, but was able to solve it... I describe how here:
http://www.brendanloy.com/2007/02/wordpress-21-upgrade-problems.html
Basically, to solve the comment_count issue, I ran the following SQL query in phpMyAdmin:
UPDATE wp_posts SET comment_count = (SELECT COUNT(comment_post_id) FROM wp_comments WHERE wp_posts.id = wp_comments.comment_post_id)
My database has 129 categories, 14,461 posts and 107,302 comments, and the query ran almost instantaneously, so speed should not be an issue.
As for categories, that can be solved without delving into phpMyAdmin. In the WordPress interface, simply create a new post and check off every single category. Save the post. Voila, you’re done. All the categories will update their category_count field. Admittedly, it was a little annoying to manually check off 129 categories, but it was doable, and it solved the problem in one fell swoop.