Как на английском будет звучать предложение:

1. The patient was asleep (was sleeping) when the ambulance arrived. (Past Simple/Continuous + Past Simple)
2. He('d) finished his work before it started to rain. (Past Perfect/Simple + Past Simple)

The patient was sleeping when the ambulance came/ arrived
He had finished the work before it started to rain.

The patient was sleeping when the ambulance arrived - past continious и past simple
He had finished working before it started raining - pas perfect и past simple

The patient was sleeping when the ambulance arrived
He had finished the work before it started to rain.

А насчет различий между before it started to rain и before it rained в этом предложении, то before it rained - это тоже правильный вариант с точки зрения грамматики, но для носителей языка it started to rain привычней. Потом под "пошел дождь" подразумевается, что он начался.

Подобный вопрос был задан год с лишним на форуме wordreference.com. Там пользователь спрашивал можно ли сказать "he had cycled before it rained". Американец ответил, что этот вариант можно считать правильным, но лучше сказать He had finished cycling before it started to rain.

The patient was sleeping when the ambulance arrived.
He finished the job before the rain came.
1) past continius
2)past simple