I'm a little confused about how 이상하다 can be used with 네요 grammar. For example consider you want to talk about an strange child.
Is "이상한 아이네요!" a correct form?
Is "아이가 이상하네요!" correct?
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아이가 이상하네요
Here, we are adding the subject particle (가) to the word child (아이), and then adding the 네요 'surprise' ending onto the descriptive verb 이상하다.
이상한 아이네요
Here, there is a 'hidden' copula, '이다'. 아이네요 is basically '아이 이다' ('it is a child') with the 네요 'surprise' ending added ('아이이네요') and then shortened to '아이네요'.
이상한 is the form of 이상하다 that is most similar to an English adjective, created by using the ending ㄴ.
So they are both correct remarks - one constructed around 이상하다, one constructed around 이다.
Both are correct. The meaning of the two sentences you mentioned almost exactly corresponds to the sentences below, respectively.
Two sentences are grammatically correct, and may convey the same meaning.
But in my opinion, "아이가 이상하네요" or "아이가 이상해요" sounds more like the speaker is worried about a child's health (or mental). On the other hand, "이상한 아이네요" sounds like the speaker is annoyed by the bad behavior of the child or simply cannot understand the behavior.
I have no theoretical support for this, but I found it also applies to some other cases. "차가 이상하네" -> something is going on with this car! "이상한 차네" -> what a strange car!
The ending '-네' means:
어미
(‘이다’의 어간, 용언의 어간 또는 어미 ‘-으시-’, ‘-었-’, ‘-겠-’ 뒤에 붙어)
- 하게할 자리에 쓰여, 단순한 서술의 뜻을 나타내는 종결 어미.
자네 차례네. 집이 참 넓네. 나 지금 가네.
- 해할 자리나 혼잣말에 쓰여, 지금 깨달은 일을 서술하는 데 쓰이는 종결 어미. 흔히 감탄의 뜻이 드러난다.
우리 아이 노래도 잘 부르네! 집이 참 깨끗하네. 개나리꽃이 정말 노랗네.
Short summary:
'-네' is a final ending used for two purposes.
집이 참 넓네. The house is very large. = 집이 참 넓다.
나 지금 가네. I am going now. = 나 지금 간다.
집이 참 깨끗하네. What a clean house it is. (How clean your house is).
Your question:
Is "이상한 아이네요!" a correct form?
Yes, '-요' is an honorific auxiliary and the sentence sounds more honorific than "이상한 아이네.". As @topomorto answered, it connotes your surprise in an exclamatory sentence. It literally translates to "What a strange kid (s)he is" or "How strange the kid is."
Is "아이가 이상하네요!" correct?
Yes, again, '-요' is an honorific auxiliary. It is more honorific than "아이가 이상히네." and it literally translates to "The kid is strange." or "What a strange kid (s)he is."