18 questions

키가 몇이에요 / 키가 몇이세요?

2. 가장 좋아하는 색은 뭐예요?  // good
What's your favorite color
(이 중) 마음에 드는 색은 뭐예요? // asking to pick one from a small selection since 마음에 들다 is for something very specific/close
What's favorite/What color do you like?

3. 가장 좋아하는 교과는 뭐예요?
What's your favorite school subject?

It sounds too formal/academic. Replace 교과 with 과목

Also consider replacing 는 with 이, depending on the context.

4. 배우고 싶다는 것은 뭐예요? / 배우고 싶은 것은 뭐예요?
What do you want to learn?

The latter is correct.

The former means "what did he/she say he/she wants to learn?"

5. 제일 좋아하는 인용구는 뭐예요?
What's your favorite quote?

Again, 인용구 is for something specific/close, for cases where you have the original full text in front of you. (or on the shelf)

Replace it with 격언

6. 제일 좋아하는 (갈) 곳은 뭐예요?  // sounds awkward
What's your favorite place (to go to)?

가장 좋아하는 여행지는 어디예요?

7. (당신이) 들었던 가장 최근(의) 노래는 뭐예요?
What is the last song/the most recent song you listened to?

가장 최근 isn't very logical since "최" is already "most"

근 means "close", and 최근 almost always refers to the point of the time when someone did something.

The whole sentence sounds like "which song did you listen to most recently". You should replace 최근 with 최신 where "신" means "new".

요즘 들은 최신곡이 뭐예요?

요즘: lately

곡: song

들었던 is incorrect since you most probably didn't stop listening to songs altogether.

8. 눈빛이 뭐예요? / 눈이 무슨 색이죠?
What's your eye color? / What color are you eyes?

The latter is correct. "뭐" has much less use than "what" in English

9. 어떤 더 낫을까요? 치마 또는 청바지?
Which is better(preferable/superior)? Skirt or jeans?

어떤게 더 나을까요? 치마? 아니면 청바지?

또는 doesn't suit very well to a "either or" situation like this.

10. 머리색깔/머리카락 색/머리 색깔은 뭐죠?
What's your hair color?

머리가 무슨 색이죠?

머리 색이 어떻게 돼죠?

11. 가장 최근(에?) 읽었던 책은 뭐예요?
What's the most recent book you read?

You are asking about the book the other person read most recently.

요즘 읽은 최신작이 뭐예요?

What's the book you read published most recently?

작: piece of work (artistic)