I have the possibility to organize a special track/session at a small computer science conference, however at the moment I'm in the last year of my PhD study.
The special session includes an introduction presentation (related to an editorial) by the organizer (me in this case), some accepted papers presentations, and an "Open Discussion and Closing Remarks" chaired by me as well.
Although I believe to have enough expertise/general knowledge on the topic of the session, do you think it'd be appropriate for me as a PhD student to do this? Or is everyone expecting a doctorate or a professor to organize such sessions?
I have done so, and the participants weren't troubled by it at all. As a PhD student you are supposed to become an expert on your (very narrow) topic, so near the end you are supposed to know more than most about that one topic than others in your field.
Susceptability to status differs quite a bit from country to country and discipline to discipline, so whether this works in your field depends. I am Dutch, but the conference was an international one, and my discipline is sociology.