The rain is coming down. (But, I would probably just say "It's raining.") The snow is coming down. (But, I would probably just say "It's snowing.") The last two sentences are perfect.

The first two sentences require the definite article (the) because they are talking about that rain and that snow in particular. It is specific to that rain and to that snow so the sentences should be: "The rain is really coming down." and "The snow is really coming down."

Add THE to the first two and you've got it. To be clear, all those sentences mean that it's raining (or snowing) hard. Not just a little. You probably already know that!