Learn Russian: Свирепый тигр ревел, когда начался ураган. - The fierce tiger roared when the hurricane began.

This lesson is about the metaphorical Instrumental, but I don't see it here. How could we rewrite this to say the hurricane roared like a fierce tiger?

You would make the hurricane the subject and put the tiger in the Instrumental case: Ураган ревел свирепым тигром. This is a classic way to use the Instrumental to show that an action resembles something else, which is exactly what your current lesson is teaching!

Why do we use the imperfective verb ревел but the perfective начался?

This is a classic 'scene-setting' contrast. The imperfective ревел describes an ongoing, continuous background action (the tiger was roaring). The perfective начался describes a sudden, completed event that happened during that background action.

Why does начался have the reflexive -ся ending?

The non-reflexive verb начать requires a direct object, meaning someone starts something. Because the hurricane isn't starting something else, but simply starting itself (intransitive), Russian requires the reflexive form начался.

Could I say когда ураган начался instead of когда начался ураган?

Yes, you could, but Russian often puts the subject at the end of a clause to introduce it as new or dramatic information. Saying начался ураган builds a little suspense and emphasizes the arrival of the hurricane.