bi on da-eume geonjodaee neoreo dun sugeoni jal an mallaseo gaseupgineun an kyeosseo.

Both are possible. In casual Korean, 가/이 is often omitted when the meaning is clear from context.

So:

• 비가 온 다음에 = the full, explicit form
• 비 온 다음에 = a more natural, conversational shortening

Here it still means after it rained.

Also, 온 is the noun-modifying form of 오다 in the past, so it is literally something like the following time after rain came.

-ㄴ/은 다음에 means after doing..., once ... has happened, or afterward.

So:

• 오다 → 온 다음에 = after coming
• 비 온 다음에 = after it rained

It is a very common pattern:

• 밥 먹은 다음에 = after eating
• 숙제한 다음에 = after doing homework

In this sentence, it sets the time/background for the rest of the sentence.

It is one long noun phrase modifying 수건.

Break it down like this:

• 건조대에 = on the drying rack
• 널어 둔 = hung up and left that way
• 수건 = towel

So the whole phrase means:

the towel that had been hung on the drying rack

This is very normal in Korean: a lot of information comes before the noun it describes.

널다 simply means to hang something out, especially laundry.

• 수건을 널다 = to hang out a towel

널어 두다 adds the idea of doing it and leaving it in that state.

• 수건을 널어 두다 = to hang the towel up and leave it there

So in this sentence, 널어 둔 수건 suggests the towel was not just hung for a moment; it had been hung there and left to dry.

Because 둔 is the past/adnominal form of 두다, used before a noun.

• 두다 = to put, leave, keep
• 둔 = that was left/kept

So:

• 널어 두다 = to hang and leave
• 널어 둔 수건 = the towel that was hung and left there

This is a very common structure in Korean when describing a noun with a completed action.

Because 수건 is the subject of 잘 안 말라서.

What didn’t dry well?
The towel didn’t dry well.

So even though there is a long description before 수건, the main noun is still 수건, and 이/가 marks it as the subject of the following verb.

It means because it didn’t dry well or because it wouldn’t dry properly.

Breakdown:

• 잘 = well
• 안 = not
• 마르다 = to dry
• -아서/어서 = because / so

So 잘 안 마르다 is a common expression meaning:

• to not dry well
• to not dry easily
• to stay damp

The 잘 does not mean the drying was good. In negative expressions like this, 잘 안 V often means doesn’t V easily/properly.

Because 마르다 is a 르 irregular verb.

When a vowel-initial ending is added, the form changes:

• 마르다
• 말라
• 말라서

So:

• 마르다
  • -아서 → 말라서

This is the same kind of change you see in:

• 모르다 → 몰라요
• 고르다 → 골라요

Using 는 gives a topic or contrastive feeling.

• 가습기를 안 켰어 = I didn’t turn on the humidifier.
  
  • more neutral as a direct object statement
• 가습기는 안 켰어 = As for the humidifier, I didn’t turn it on.
  
  • more topical, and often slightly contrastive

In natural Korean, X는 안 V is very common when saying what was or wasn’t done. It can imply something like:

• at least the humidifier, I didn’t turn on
• or simply regarding the humidifier, I didn’t turn it on

The contrast may be strong or weak depending on context.

Here -서 gives the reason or cause:

• 잘 안 말라서 = because it didn’t dry well

So the logic of the sentence is:

1. It rained.
2. The towel hanging on the drying rack didn’t dry well.
3. Therefore, I didn’t turn on the humidifier.

In this sentence, -서 links the towel not drying properly to the decision not to turn on the humidifier.

안 켰어 is casual, intimate speech.

It would be used with:

• friends
• family
• someone younger
• very casual conversation

More polite versions would be:

• 안 켰어요 = polite casual
• 안 켰습니다 = formal

Also, 켰어 is the past form of 켜다 (to turn on).

Yes, it is completely normal Korean word order.

Korean often places a lot of descriptive information before the noun and before the final verb. The sentence structure is roughly:

• 비 온 다음에 = after it rained
• 건조대에 널어 둔 수건이 = the towel that had been hung on the drying rack
• 잘 안 말라서 = because it didn’t dry well
• 가습기는 안 켰어 = I didn’t turn on the humidifier

So even though it feels long to an English speaker, this is a very natural Korean way to package information.