I can help with #1 and #3. The ending -(으)ㄴ가요 is used only in interrogative sentences and it softens the question, making it more polite than 이다+요. It is attached to 이다 in your example sentence. #3. This is not the object particle. This is the future tense ㄹ particle, as in 먹을 거예요. It is used here as a modifier to the verb 가르치다 because the teaching will take place in the future. In the English sentence, the future possibility of "the message we want to teach to our kids" is implied through the word "want" but in Korean it is given through the ㄹ. Where did you see/hear this sentence? If it was texted to you, it's possible they made a mistake. I'm not expert enough to be sure about your second question, but my instinct is that it should be the topic particle 은 attached to 이것은 to make the sentence clearer.

이것이 우리가 아이들에게 가르치길 원하는 메시지인가? Is this the message that we want to teach our kids? 1) Noun + 인가(요) is a way to ask questions. How is it different from 이에요/예요? You've been heard the six speech levels of Korean. Note 하게체 and 해요체, please. Followings are each Interrogatives of the six level of speech. 이것이 책이다. This is a book. 합쇼체: 이것이 책이-ㅂ니까? (이것이 책입니다) 해요체: 이것이 책이-어요? or 책이-에요? (이것이 책이어요, 책이에요) 하오체: 이것이 책이-오? or 책이-요? (이것이 책이오) 하게체: 이것이 책이-ㄴ가? (이것이 책일세 or 책이네) 해체: 이것이 책이-야? or 책이-니? or 책이지? (이것이 책이야 or 책이지) 해라체: 이것이 책이-냐? (이것이 책이니라) By adding "요" to the sentence endings of 해체, It is changed into 해요체 except "-니". 이것이 책인가(요)? 이것이 책이지(요)? 이것이 책이야(요)?, (Now, this form is used only North Korean) 2) There are two words with subject marking particles. 이것이 is the subject of the main sentence: "Is THIS the message?" And 우리가 is the subject of the clause: "that WE want to teach our kids". Do I have that right? Yes, Right. 3) I think the word 가르치길 is the noun form of 가르치다 (가르치기) with the object particle -ㄹ attached. I don't usually see the object particle attached to noun forms like this. Is it normal? Yes, It is usually used in such as form. 무엇을 원하다, Noun or Noun form of verb + 원하다. 무엇을 하고 싶다. Do you want a coffee? 커피를 원하니? Do you want drinking a coffee? 커피(를) 마시기를/마시길 원하니? (I don't know whether this sentence is correct in English grammar or not ) Do you want to drink a coffee? 커피를 마시고 싶니? Please compare the difference of these two sentences, between 먹기 and 먹는 것. 먹다, 먹-기, 무엇이 힘들다. 뭐가 힘들다. 이 과자는 너무 딱딱해서, 먹기(가) 힘들다. or 이 과자는 너무 딱딱해서, 먹는 것이 힘들다.