Heads-up: this is a quick partial answer — please accept a more detailed one.
It’s about omitting postpositions/markers.
- All of the sentences sound good and natural to me.
- Grammatically, both are identical. Not this one, but there are some cases where fuller forms are not idiomatic and sounds more awkward. E.g. “너 심부름 가니?” sounds much better than “너 심부름을 가니?”. The exact reasons and rules ⸺ I don’t know.
- Although those without postpositions are more common and seemingly preferred by native speakers, it is not that you can drop any postpositions and make sense. From a related paper:
Omitting agentive postpositions (subject markers)
TL; DR: if a subject is specific and always interpreted as a subject (“doesn’t misinterpreted as an object”) even without a subject marker, you can drop the marker.
‘너 심부름 가니?/저 개 아주 사납다/OOO의 소설 재미있더라.’와 같이, 대명사나 그 지시 대상이 특정적인, 한정적 표현의 명사가 주어 자리에 올 때 주어가 주격 조사 없이 쓰일 수 있다. When a subject is a pronoun or a specific noun, you can omit the agentive postposition (subject marker).
- ‘너 심부름 가니?’ “Are you going on an errand?”
- ‘저 개 아주 사납다.’ “That dog is so aggressive.”
- ‘……의 소설 재미있더라.’ “……’s novel was interesting to read.”
하지만 ‘개 아주 사납구나./소설 재미있더라.’와 같이, 주어 자리에 오는 명사가 비한정적일 때에 주격 조사가 쓰이지 않으면 비문(非文)이 된다. If you don’t put an agentive postposition (subject marker) on a subject that is not specific, the sentence becomes ungrammatical.
또한 ‘그 여자 무얼 주려고 하는데.’와 같이 주격 조사가 없이 쓰인 체언이 주어 이외의 것(‘그 여자가’ 혹은 ‘그 여자한테’)으로 혼동될 우려가 있을 때는 주어 자리에 오는 체언이 비록 한정적이라도 주격 조사가 쓰여야 한다. If a subject without an agentive postposition (subject marker) could be misinterpreted as a non-subject, you have to keep the agentive postposition (subject marker), whether the subject is specific or not.
- The ambiguous sentence ‘그 여자 무얼 주려고 하는데.’ (“The girl; about to give something.”) can be interpreted as:
- ‘그 여자(가)[subject “She”] (누구에게)[implied object “to someone”] 무얼 주려고 하는데.’ “She(’s) about to give something (to someone).”
- ‘(내가)[implied subject “I”] 그 여자(한테)[object “to her”] 무얼 주려고 하는데.’ “(I’m) about to give something (to) her.”
⸻ Gisim Nam, ‘현대 국어 통사론’ (“The syntax of modern Korean”), published by 태학사 (Taehaksa), in 2001.
Some other postpositions you could omit (couldn’t find the documented rules though):
- 목적격 조사Accusative postposition (object marker) “을”/“를”
- 속격 조사Genitive postposition “’s” “의”
- 향격 조사Lative postpostion “to” “에”/“로”
- and possibly more.