Learn Hardcore Turkish: Misafirim gelmeden çarşafları yıkayıp serdim. - Before my guest arrived, I washed the bedsheets and laid them out.

As a standalone predicate, misafirim means “I am a guest.”
Here it precedes gelmeden, so it’s a possessed noun meaning “my guest” and serves as the subject of the subordinate action.
A quick test: if the word is doing something (e.g., misafirim gelmedi), it must mean “my guest,” not “I am a guest.”
You can add benim for emphasis: benim misafirim = “my guest,” but benim is optional.

Why does gelmeden have the negative -me if the meaning is “before (my guest) came,” not “without coming”?

The converb -meden/-madan (negative + ablative) commonly expresses “before V-ing” relative to the main clause’s time.
It can also mean “without V-ing,” depending on context. In this sentence, the natural reading is temporal: “before.”

Can I say gelmeden önce or gelmezden önce instead? Any nuance?

gelmeden önce is fully natural and makes the “before” meaning explicit.
gelmezden önce is more formal/literary; you’ll see it in writing more than speech.
gelmeden alone is also fine; adding önce is often stylistic or to avoid any potential “without” reading.

Could gelmeden here mean “without my guest coming”? How would I force that reading?