How would you use "only" or "alone" to denote whether something happened exclusively in one place? For example "It happened only in the United States" or "It happened in the United States alone". The second one "seems" right, but is the first one wrong? How would I explain this to a non-native speaker?
With a focusing modifier like "only", the only practical rule in writing is to place it sensibly so as not to cause confusion. "Only" usually precedes its focus, and it can often be placed adjacent to it, as in "We found only one error", in which case "only" is modifying the NP "one error". But if the focus is contained within a verb phrase, as it is in that example, then non-adjacent is usually fine, as in "We only found one error", in which case "only" is now modifying the whole verb phrase.
In your example, placing "only" adjacent and before "United States" is clearly not an option, so it's fine to place it before the preposition "in", in which case it's modifying the whole preposition phrase. Also, there is the option here to place it immediately after the focus, as in "It happened in the United States only", though this might be the least preferred option.
In the case of "alone", things are a tad different since it can only be placed after its focus (which incidentally can only be a noun phrase), so "It happened in the United States alone" is the only option.