Does Time Out have breaks while the computer is idle?

No, Time Out automatically pauses the breaks when you aren't using your computer. So when the screensaver is active, or the computer is otherwise sitting there without being used, Time Out won't do anything.

When you return to your Mac, if you've been away for longer than the interval set in the preferences, Time Out will reset the break timers, so the next breaks will be the full work interval from that time. You can change this amount or disable this feature in the preferences.