When a countdown expires, there is a delay before the tournament starts, just like the one between rounds. By default it is 4 seconds, but you can change this on the Preferences tab.
So, what is actually happening is the clock is pausing at 0 seconds left in the countdown. Until that 4-second delay passes, the tournament hasn't started, and technically it's still in the countdown period. So when you start the clock again (unpause the clock), the 4-second delay commences (again, technically it's still in the countdown period) and when it's done, the tournament starts. This is when the "Countdown a specific length of time" option is selected. So this is really working as designed.
Now, when you select "Countdown until a specific time of day", that's a little more interesting. In this mode, you can't "pause" the countdown clock, because, well, that just doesn't make sense. You're counting down to a specific time, and you can't pause real time, so the countdown clock will continue to tick whether or not you chose to pause it or not. The software doesn't allow you to pause the clock during this type of countdown, but the event is overriding this (making this a bug). The clock reflects this in that the countdown time begins to count up (because you've passed the time of day that it was supposed to end). Since the clock is effectively "paused", the tournament never begins, and since you can't use the spacebar (or other hotkey) to toggle the clock in this type of countdown, you can't start the tournament that way. You CAN start the tournament by pressing the "Start Tournament" button on the Controls tab.
I'm not really sure how to handle this. I suppose the event code might have to be much smarter and see that the event is attempting to pause the clock during a TOD countdown, and that the countdown is at 0 (in essence), so it should start the tournament and then pause the clock. Maybe. Have to give this some thought.