For what it's worth, I think that a lot of these sentences involving leesik should be translated into English without the word "down". It's redundant (since "fall" naturally indicates downward motion anyway), and not only that, "fall down" is actually a phrasal verb in English with a pretty specific meaning - "collapse" or "fall to the ground" - that is different from simply "fall + downward direction," and which may not be wanted.
I think both should be acceptable answers even though technically "down" is redundant (though arguably one can "fall to the side", "fall over" and although the overall direction is downwards they do convey a subtly different meaning). Using "fall down" instead of plain "fall" is very common (where I am from at least!), and also is commonly encountered in literary / poetic use. I'm interested to know if the directional "le" is redundant in Hungarian? Can you say "Mi esik a tetőről"?
No, it's not redundant, it's purpose is more to give a perfective aspect to the verb than to express the direction. If something "leesik", it will end up on the ground. "Mi esik a tetőről?" could be translated as "What is falling from the roof?" It's in the process of falling, it's still in the air, if you have good reflexes, you might catch it before it collides with the ground.
"Eső" is rain (literally: the falling) "Eső esik" or "esik az eső" Means: "it is raining" (Literally: the falling is falling) As a continuous state, I think, "esik" is reserved for rain and perhaps waterfalls and alike.
Everything else should hit the ground sooner or later, so it needs "le-"
Now if you talk about "washing down" you can say (le-mosódik), or you'll need to describe the liquid state and say it is "flowing" (le-folyik) or "pouring" (le-zuhan) down.