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.