I think 잠잡니다 is a simplified version of 잠을 잡니다

The 을 / 를 is called the object marker and marks the object of a sentence. In short sentences like these, it is not mandatory and can be omitted.

I would think, as a beginner, that both are correct. I think using 는 over 가 would be a matter of conversational context. Looking forward to a solid and knowledgeable answer. Good question.

Yeah, I was thinking the same thing

는 is a topic marker, marking what the sentence is about. 가 is a subject marker, marking the subject of a sentence (the word that is being acted upon by the verb). There can be cases, like this one, that both of these markers can refer to the same thing. But since the sentence "아이는 잠잡니다" is simple with only the noun (child) and verb (sleeps), it is not necessary to mark 아이 with 가 because it is obvious that child is the subject of the sentence, there is no other possible subject. However if you wanted to put extra emphasis on the child I believe you could use 가 in place of 는.

I hope this clarified. If anyone else has anything to add feel free.

Yes, it does. 잠 means sleep. It is an expression like "die a death" or "live a life" in English, but 잠자다, 춤추다 etc. are very commonly said in Korean.

Yes, 자요 is informal polite, whereas 잡니다 is formal polite.

I think it would be fine, it's just a bit less formal than 잠니다.

는 is a topic marker, marking what the sentence is about. 가 is a subject marker, marking the subject of a sentence (the word that is being acted upon by the verb). There can be cases, like this one, that both of these markers can refer to the same thing. But since the sentence "아이는 잠잡니다" is simple with only the noun (child) and verb (sleeps), it is not necessary to mark 아이 with 가 because it is obvious that child is the subject of the sentence, there is no other possible subject. However if you wanted to put extra emphasis on the child I believe you could use 가 in place of 는.

I hope this clarified.