In that case, does "I fell in a hole" imply that the hole is horizontal and going sideways?

Is there a similar implication for stairs when people fall down them?

If I'm walking around in a hole and stumble, I've also fallen in a hole (but not into/down)

Yeah. I feel like fell down implies you travelled some not insignifcant distance while falling.

All language is like this. It's a large part of why communication is actually very difficult.

Grammatically, no, because "or" is a coordinating conjunction (for, and, nor, but, or, yet, so), and coordinating conjunctions are a way to join two independent clauses, like a semicolon. They are used after an independent clause and a comma, and they are followed by another coordinating conjunction.

Here's two independent clauses: I got scratched by a cat. I'm sad.

Here's a way to join them with a semicolon: I got scratched by a cat; I'm sad. The semicolon replaces the period.

Here's a way to join them with a coordinating conjunction: I got scratched by a cat, so I'm sad. The , so replaces the semicolon/period.

Note that I got scratched by a cat so, I'm sad is incorrect, because to join two independent clauses, you're supposed to put the comma first and then the conjunction, in that order. Colloquially, people will often omit the comma entirely, to reflect pronunciation I guess. But as far as I can tell, people don't generally pronounce a pause between the coordinating conjunction and the following independent clause, so they don't put a comma there either.

No I don't think it's necessary at all, unless the author wanted to emphasise a pause there