"It rained last week."
Yes, I agree with you. "下雨" means "to rain" (as a verb).
下了雨 seems very odd
No. 'Le' is used differently in different places.
There are many uses; but most popular are:
It is used directly after the verb to indicate a past event
It is used after a full sentence to indicate that some status has changed.
This case is the first one. Since 'xia yu' is actually a mixture of xia(to pour; verb) and yu(rain; noun), le has to be directly after xia