Learn Hardcore Korean: seorap ane osdo jeongrihaesseoyo. - I organized clothes inside the drawer as well.

서랍 means “drawer.”
안에 is a compound of 안 (“inside”) + -에 (location particle), so 서랍 안에 = “inside the drawer.”
If you say 서랍에, it usually means “at/to the drawer,” which can be ambiguous (you might be standing by it). 안에 specifically pinpoints “within” the drawer.

Why is it spelled 안에 and not 않에?

안 (inside) and 않 (negation root of 않다, “not do”) are different words.
안에 uses 안 (“inside”).
않에 would combine the negation root 않 with -에, but that doesn’t make sense because you don’t say “not inside.”

What role does 도 play in 옷도 정리했어요?

도 means “also” or “too.”
When attached to 옷, 옷도 means “(the) clothes, too.”
It implies you organized clothes in addition to something else (for example, shoes or books).

Could I instead say 서랍 안에도 옷을 정리했어요? How would that change the nuance?

Yes, you can.
서랍 안에도 (“also inside the drawer”) emphasizes the location as an additional place you organized—“I also organized clothes in the drawer (besides other places).”
In the original, 옷도 focuses on adding “clothes” to a list of items you organized.

Why is there no subject like 제가 or 저는 in the sentence?

Korean often omits the subject when it is obvious from context.
Here, the speaker (I) is understood.
If you want to be explicit or contrast, you can say 저는 서랍 안에 옷도 정리했어요 (“As for me, I also organized clothes in the drawer.”).

What is the nuance of 정리했어요 and what are its other common forms?

Is the word order 서랍 안에 옷도 정리했어요 fixed, or can I say 옷도 서랍 안에 정리했어요?

Korean word order is relatively flexible because particles mark each word’s role.
옷도 서랍 안에 정리했어요 is also correct and means the same thing—“I also organized clothes inside the drawer.”
Starting with 서랍 안에 just sets the scene on location first, but neither order is ungrammatical.