When I reply to a thread that I'm already subscribed to, do I need to tick the box that says "subscribe to this thread" in order to continue the subscription?
It depends on how you have set your subscription preferences. (Accessible from your subscription page via the "Preferences" link.) See the section "Subscribe by default when I post or reply to..."
(I suggest ensuring that you become automatically subscribed to a thread which you comment in. You can always manually override this for a given thread if you don't want to stay subscribed to it.)