Learn Russian: Лист падает.

Not all independent actions need -ся. The -ся ending is mostly added to verbs that usually do something to an object (like to wash or to open) to show the subject is doing the action to itself or that the action is happening by itself. The base verb падать (to fall) is naturally an action that happens on its own and never takes a direct object, so it does not need a reflexive ending.

Does Лист падает translate to "The leaf falls" or "The leaf is falling"?

It translates to both! Russian only has one present tense. Depending on the context, падает acts as either the simple present (falls in general) or the present continuous (is falling right now).

I've seen the word лист used for a sheet of paper. Does it also mean a tree leaf?

Yes, it means both. The word лист is used for a leaf on a plant as well as a sheet of paper. Here, the action of falling naturally hints that it is a tree leaf, though a piece of paper can fall too! Context will always guide you.

If I wanted to say "The leaf fell" in the past tense, how would I know which gender ending to use for the verb?

You can tell the gender of лист by looking at its last letter. Because it ends in a hard consonant (т), it is a masculine noun. To make it past tense, you would use the masculine past tense form of the verb, which is Лист падал.