It should renew unless there's a hold on it or you've renewed it the max allowed times. And yes, as long as it is in that drop box tonight, it will count towards today's date. (I know both through personal experience.)
You may want to try and renew online 1st. Not sure with dvds if it will work. If it doesnt, I would put it in the drop box and you can always call the next day. The librarians are very accommodating at that branch.
Thanks! Your answer is awaiting moderation.