Learn Hardcore Turkish: Yağmur dinince yağmurluğu katlayıp çantama koydum, kapüşonu da kuruttum. - When the rain stopped, I folded the raincoat and put it in my bag; I also dried the hood.

It’s the temporal suffix -ince/-ınca/-ünce/-unca, meaning “when/once …”. It forms an adverbial time clause.
The verb base here is dinmek “to stop/abate (rain, wind, noise)”, not dinlemek “to listen”.
So yağmur dinince = “when the rain stops.”
Vowel harmony picks the right variant of the suffix: din-
- -ince → dinince.

Could I say yağmur durunca, yağmur durduğunda, or yağmur dindikten sonra instead? Any nuance?

yağmur durunca and yağmur dinince are near-synonyms; dinmek is a bit more idiomatic for rain easing/stopping, durmak is more general “to stop.”
yağmur durduğunda (with -DIK + 3sg poss + loc) is a bit more formal/literary: “when the rain stopped.”
yağmur dindikten sonra is “after the rain stopped,” explicitly emphasizing “after.”

How is yağmurluğu formed, and why not simply yağmurluk?

yağmur-luk = “raincoat.”
As a definite direct object, it takes the accusative -ı/-i/-u/-ü → yağmurluk + u → yağmurluğu. The final k softens to ğ.
Bare yağmurluk would read as “a raincoat” (indefinite) in object position; yağmurluğu is “the raincoat.”

Shouldn’t it be yağmurluğumu if it’s “my raincoat”?

If you want to explicitly say “my raincoat” as a definite object, use 1st person possessive + accusative: yağmurluğum → yağmurluğumu.
The sentence as given says “the raincoat” (definite) without marking whose it is. That’s perfectly fine; context often makes it obvious it’s the speaker’s. Use yağmurluğumu only if you want to make the possession explicit.

Why is it kapüşonu and not kapüşonunu?

kapüşonu is the accusative of “hood” as a definite object: “the hood.”
kapüşonunu is 3rd-person possessed + accusative: “its hood” (e.g., the hood of the raincoat).
In context, Turkish can use kapüşonu to mean “the hood (of that raincoat we’re talking about),” relying on context. If you want to be explicit about the ownership, say yağmurluğun kapüşonunu or just kapüşonunu (referring back to the raincoat).

What does -ip in katlayıp do?

-ip links verbs with the same subject to show sequential actions, often “do X and then do Y.”
yağmurluğu katlayıp çantama koydum = “I folded the raincoat and (then) put it in my bag.” The object is understood to carry over to the next verb unless changed.

Is there a difference between -ip (katlayıp) and -erek (katlayarak)?

-ip is a neutral, light way to chain actions, typically sequential.
-erek/-arak often means “by doing” and implies manner/simultaneity. Katlayarak çantama koydum sounds odd because you can’t really “put it in the bag by folding it (at the same time).” -ip is the natural choice here.

Why çantama and not çantaya?

What does da in kapüşonu da mean, and how is it written?

da/de (written separately) is the additive clitic meaning “also/too.” Here: “I also dried the hood.”
It obeys vowel harmony: after a back vowel use da, after a front vowel use de. Hence kapüşonu da.
Don’t confuse this with the locative suffix -da/-de, which attaches to the word (e.g., evde “at home”).

Why is there a comma before kapüşonu da kuruttum? Could I add ve?

You’re coordinating two clauses: “I folded…and put…,” and “I also dried the hood.” A comma is natural; you could also use ve:
… çantama koydum ve kapüşonu da kuruttum.
Both are fine.

How is the object handled across the two verbs in yağmurluğu katlayıp çantama koydum?

The explicit object yağmurluğu belongs to katlayıp. For koydum, Turkish simply carries over the same understood object (no need to repeat onu “it”). Repeating the object would sound heavy.

How do you pronounce ğ in yağmurluğu?

Turkish ğ (yumuşak g) is not a hard “g.” It typically lengthens the preceding vowel or creates a light glide.
yağmurluğu is roughly “yaamurluu” (smoothly linking the vowels), not “yag-mur-lug-u.”

Why isn’t the subject ben stated?

Turkish is pro-drop: verb endings show the subject.
koydum / kuruttum both have -m (1sg), so ben is unnecessary unless you want emphasis.

Is kuruttum related to kurumak?

Yes. kurumak = “to dry (intransitive: something dries).”
kurutmak = “to dry something (transitive).”
kuruttum = “I dried (it).” The double t is normal in the stem kurut- plus past -tu plus 1sg -m.