My feeling is that you don’t have to repeat the preposition if it’s the same preposition, as in 1. But if the prepositions are different, as in 2, then they are both required.

Proper way to do it is “Are you in London or New York?” and “Are you at home or in the office?”, note that you can also use the preposition “at” to say “at the office”.

You could say “where’s the remote? Under the bed, table, couch?” And it would imply that “under” goes with all three. But saying “under” with all of them wouldn’t be wrong either; one is just quicker. Both are fine. If the prepositions are different, then they should both be stated, otherwise the first preposition listed is implied to go with all the nouns that follow.

You don’t need to repeat the preposition if it is the same. So “Where are you – in London or New York?” is correct. Also the prepositions do not need to be capitalised in that sentence.

Personally, I would recast each of those sentences. The world isn’t binary: “London or New York” and “office or home”. So I would ask the questions as:

“Are you in London, New York or somewhere else now?”
“Are you still in the office or have you gone home or somewhere else?”

Even better, I might not make any supposition at all about where the person is, and simply ask: “Where are you now?”

In any case, any of your examples would be perfectly fine. All we can tell you is “this is how we speak English in this area”. People in other parts of the world speak the same language completely differently to our ways, and it makes perfect sense among them. Speakers in England think that Americans speak the language badly, but we now outnumber them. So unless we make actual errors in usage, “differences” are just part of the evolution of the language. Only the French seem to think that they can control the growth of their own language… and of course some Germans who think that everyone should speak English as they do.

For example, on a construction project in India we might get a report from the (Indian) construction manager that “There are twenty-five number welders on the project.” That sentence grates horribly on my ears, but it seems that every native Indian who speaks English speaks about quantities in exactly this way. It’s somewhat maddening, somewhat endearing, and near universal. I’m not about to tell an entire nation containing more English speakers than the United States and the UK combined “You’re doing it wrong.” They do it differently, is all.

Don’t worry overmuch about details as fine as the one you asked about. Fluency will come with time, experience, practice and listening to how others speak and write.