What month do you start school?
When did it happen? What year, what month, what day, what time, what second was it? How many times did Earth spin when it happened?
What month were you born in, Ms. Tanaka?
You have what month - as in: what month is it? It's １０月, you fool. Get a job, hippy.
The readings are the readings you learned with the kanji. 何 is technically the kun'yomi, but it feels pretty on'yomi-ish if you ask me. The 月 uses the on'yomi, though.