I think the most natural way to construct this idea would be something like 'London is the capital of which country...' On something like a game show it would be probably be expressed like that because, as has been mentioned, the question constructions are not particularly elegant here.

Yes, it is OK. It's perfectly correct. But it's not an everyday construction, and I don't blame you for thinking it sounds weird. Sometimes we have no choice but to create awkward-sounding sentences like this for tests, quizzes and so on. Colloquially, a native speaker might go for the preposition-at-the-end option, and say something like, 'Where's London the capital of?' or 'What country's London the capital of?' which are equally awkward-sounding, but a bit shorter and more informal. Sometimes there just isn't a neat way of saying things!

"What country is London the capital city of ? "

It's correct but it sounds weird because in conversation, you will often hear: "Which country is London the capital of?" This is technically incorrect because it puts the preposition at the end of the sentence instead of next to "which". But you will hear it said this way much more often than the technically correct construction.

Thank you for your help)