That's not really part of modern standard English. I'm sure you could find it in dialects or old texts, but in modern English, the normal way of creating questions is by using the auxiliary do – unless the question starts with a question word.
Compare: Who owns the chain? this is normal because who is a question word.
If you start the sentence with to whom instead, that is not a question word, it's a prepositional phrase.
Another example: Who wrote the letter? but To whom did you send the letter? not To whom sent you the letter.
Yes and no. In spoken English you can get rid of the do/did by intonation/pitching your voice higher at the end of the sentence, turning any statement into a question. In written English you just add a question mark. "The milk is in the fridge?" "The chain belongs to you?" "The chain belongs to whom?" This structure is very common in spoken English or written English with dialog.
Scroll down to the last example http://m.wikihow.com/Change-a-Statement-to-Question
'To whom belongs the chain' isn't an example of that type of construction though, it would be as you said 'The chain belongs to you?'.
For the more colloquial question construction, we have the same thing in Swedish. If we aren't sure intonation alone will convey it, we often add så for clarity. So we can say Kedjan tillhör dig? or Så kedjan tillhör dig? Or we add some sort of tag at the end, most commonly va: Kedjan är din, va? (there are a few other options, too).
This kind of question is very common in spoken Swedish.
jwbards, not only does your suggestion miss the "belongs", it also translate "vem" as "whose", which is not correct. It means "who".
It is true that your proposed sentence is "situationally equivalent" to the DL English sentence here, in the sense that a speaker might utter it in the same situation that the DL sentence would be said.
But the fact remains that that is not what was said here. Our job is to translate as accurately as possible what we are given.