Why does Miki keep pulling out a stopwatch?
Yes, it is a stopwatch. Ikuhara Kunihiko (Director/Producer, Storyboards, Original Concept) answered this question in an Animage interview with Be-Papas, the animation studio that created Utena:
Translation by a fan:

Animage magazine: The next question came up most often.
Question: What does Miki measure with his stopwatch? (reader from Hokkaido)
Ikuhara: Let’s just say that it’s connected with the structure of the world.
All: (laugh).
Enokido Yoji (Animation Composition, Chief Screenwriter): That’s a really hard-to-understand answer.