Learn Hardcore Korean: jeomwonege pojangeul butakhaesseoyo.

It marks the recipient/target of the action—roughly “to” or “from the perspective of” someone. Here it indicates the person you directed the request to: the clerk. It’s the typical dative marker used with people/animals.

Can I replace 에게 with 한테 or 께?

한테: Yes. More colloquial/natural in speech: 점원한테.
에게: More neutral or written; still fine in speech.
께: Honorific “to.” Use when you want to be especially respectful to the recipient (e.g., a teacher, elder, customer). With a store clerk, 점원분께 can sound very polite, but everyday speech usually sticks to 한테 or 에게.

Why is 포장을 a noun here instead of using the verb 포장하다?

Korean often turns requests into “N + 을/를 부탁하다” (“ask for N”). 포장을 부탁하다 literally means “ask for wrapping/packaging.” A very common verb-based alternative is:

포장해 달라고 (점원한테) 부탁했어요 = asked (them) to wrap it for me.

Is 부탁했어요 the same as “asked a question”?

No. 부탁하다 is “to ask a favor / request.” For asking a question, you use 묻다/물어보다. So:

부탁했어요 = requested (a favor).
물어봤어요 = asked/inquired.

Would I actually say this sentence to a clerk at the counter?

As a direct request to the clerk, you’d usually say:

포장해 주세요.
포장 좀 부탁드려요.
이거 포장해 주시겠어요? Your original sentence is more like reporting to someone else what you did (“I asked the clerk…”), not what you’d say to the clerk.

What’s the difference between 부탁했어요 and 부탁드렸어요?

부탁했어요: neutral “I requested.”
부탁드렸어요: humble form; it raises the listener/recipient and sounds more polite/formal. To be extra courteous about your request to the clerk, you might say 점원분께 포장을 부탁드렸어요.

Can I drop the object marker and say 포장 부탁했어요?

Yes. In casual speech, object markers are often omitted when the meaning is clear. All are acceptable:

포장을 부탁했어요 (explicit)
포장 부탁했어요 (natural, colloquial)

Is the word order fixed? Can I say 포장을 점원에게 부탁했어요?

Word order is flexible in Korean. Both are fine:

점원에게 포장을 부탁했어요 (neutral)
포장을 점원에게 부탁했어요 (slight emphasis on what was requested) The verb normally comes at the end.

Does 포장 mean gift wrapping or takeout?

Both, depending on context:

In shops: “gift wrapping/packaging.”
In restaurants/cafés: “to-go/takeout.” You’ll often hear/ask 포장해 주세요 to mean “Please make it to go.”

Is 에게서 or 에서 correct here?

No.

에게서/한테서 = “from (someone)” as a source (e.g., 받다/배우다). Not used with 부탁하다.
에서 = “at/from (a place).” With 부탁하다, use 에게/한테/께 for the person you are asking.

How would I make the request with a verb clause instead of a noun?

Use the “V-아/어 달라고 부탁하다” pattern:

점원한테 포장해 달라고 부탁했어요. This explicitly means “asked (the clerk) to wrap it (for me).”

How do I pronounce the sentence naturally?

점원에게: jeom-won-e-ge
포장을: po-jang-eul
부탁했어요: bu-tak-hae-sseo-yo In fast speech, 부탁했어요 often sounds like “bu-ta-ke-sseo-yo” due to ㅎ assimilation.

What politeness level is -했어요, and how can I change it?

-했어요: polite informal past (everyday polite). Other options:
Casual: 부탁했어.
Formal polite: 부탁했습니다.
Extra polite/humble: 부탁드렸어요 / 부탁드렸습니다.

How can I specify what is being wrapped?

Add the item and demonstratives:

이거 포장해 주세요. (Please wrap this.)
Reporting: 점원한테 이 선물 포장해 달라고 부탁했어요. (Asked the clerk to gift-wrap this present.)

Is there a simpler, very natural way to say this in daily conversation?

Yes. Two very common choices:

Reporting: 점원한테 포장 부탁했어요.
Direct request: 포장해 주세요 or 포장 좀 부탁드려요.

Any etiquette tips for sounding polite?

Add softeners like 좀 and use -주시겠어요? or -드려요:

포장 좀 부탁드려요.
포장해 주시겠어요? These sound courteous without being stiff.