Learn Hardcore Korean: seonmureul eonje julji jeodo jal mollayo. - I don’t really know when I’ll give the gift either.

-지 turns a verb into an indirect question or embedded clause. Here, 주다 (to give) becomes the future/adnominal form 줄 (the “will give” stem) plus -지, so 줄지 means “(whether or) when (someone) will give.” Combined with 언제, 언제 줄지 literally means “when they will give.”

What does 잘 add to 잘 몰라요?

잘 normally means “well,” but with 모르다 (to not know), 잘 모르다 conveys “to not quite know” or “to have no real idea.” So 잘 몰라요 = “I really don’t know.”

Why is 저도 used here?

저도 = “I also.” The particle -도 means “also/too.” By saying 저도 잘 몰라요, the speaker indicates that they, too, don’t know (perhaps in response to someone else saying the same).

Why is 선물을 언제 줄지 placed at the beginning? Could it be moved?

Korean places subordinate clauses (like this indirect question) before the main verb 모르다. You could also say 저도 선물을 언제 줄지 잘 몰라요 or 저도 잘 몰라요, 선물을 언제 줄지, but the clearest, most natural order is 선물을 언제 줄지 저도 잘 몰라요.

Who is the implied subject of 줄지?

Korean often omits a subject when it’s clear from context. Here, it’s understood that someone else (not the speaker) will give the gift. If you needed to specify, you could add 그(가/분이) before 언제, e.g. 그분이 언제 선물을 줄지…, but omission is normal.

Can I say 잘 모르겠어요 instead of 잘 몰라요?

Yes. 잘 모르겠어요 is slightly softer or more tentative due to -겠어요, which adds speculation. Both mean “I don’t really know,” but 몰라요 is more straightforward; 모르겠어요 sounds a bit more polite or uncertain.

Why do we use 줄 instead of 주다 before -지?

When attaching -지 (or many other suffixes), you drop the dictionary ending -다 and use the verb stem. To express future/adnominal meaning you add -ㄹ, so 주다 → stem 주- + -ㄹ = 줄, then + -지 = 줄지.

How would I turn this into a question like “Do you know when they will give the gift?”

Replace 모르다 with 알다 and use a question ending. For example:
선물을 언제 줄지 아세요?
Here 아세요 is the polite form of 알다, so the sentence means “Do you know when they will give the gift?”