Could this also mean “The rain is coming”?
Well, literally, yes. 오다 means "come".
Another definition of 오다 is for rain or snow to fall down from the sky. So it's not correct to translate it as, "rain comes."
alternatively '비가 내리다' - 'the rain is coming down'
I always think, "자크 내리다."
Years ago I learned 비가와요 - what's the difference?
No difference, just a different speech level