After taking a look at some search results and at Matt Martz’ article, Separating Pings from Comments in WordPress 2.7, I still had some issues with particularly the comment count. There were also a problem with how the functions I have added/modified are being carried across in my templates (but probably are mostly in the scope of this Blueprint theme). The files that we’re mostly going to deal with are these ones: comments.php, functions.php and single.php.
Anyways, I just want to note here how I took care of them and what I ended up with.