Learn French: L'étape commence dans dix minutes. - The stage starts in ten minutes.

The preposition dans is used to indicate the amount of time until an action begins (ten minutes from now). If you used en, it would mean the action itself takes ten minutes to complete. Since we are talking about the waiting time before the start, dans is the correct choice.

I thought the French word for "stage" was le stage. Why use l'étape here?

Le stage is a false friend; it actually means an internship or a training course. To talk about a stage in a race (like the Tour de France), a phase of a project, or a step in a journey, you use l'étape. If you meant a theater stage, you would use la scène.

Why is the present tense verb commence used for an event happening in the future?

Just like in English, French frequently uses the present tense for future events that are scheduled or happening very soon. You could also use the near future (va commencer) or the simple future (commencera), but the present tense is very natural and common here.

Could we use après instead of dans to say it starts "after ten minutes"?

No, to say that an event happens a specific amount of time from now, you must use dans. The word après (after) is used to sequence events and is usually followed by a noun, like après la course (after the race).