I would say no. A heavy rain can be like a waterfall (exaggeration). (less water seems like more water) But a waterfall can never be like a heavy rain due to their relative magnitudes. This would be saying the waterfall was getting weaker and weaker (less water instead of more water).

Sorry, I can't say that it sounds natural at all. I've never heard a waterfall described that way. Normally, we just call a waterfall a waterfall. If it were loud you might say that it was 'thunderous.'

Yes, that sentence is correct. :) However, you could also add a bit more detail to the sentence, to give better accuracy for the meaning. For example: You might say "The waterfall sounded like heavy rain" (past tense) OR you could also say "The waterfall was like hearing heavy rain" (past tense) You can always change the tense depending on what time the sentence occurred in. Hope that helps :)

It's correct English. However, the flow of water in a waterfall, even a small waterfall, is much heavier and denser than the flow of rain in a rainfall, even a "heavy downpour." Furthermore, to me the sound of a waterfall is completely different from the sound of rain.