First of all, “can” is obscure for conveying the exact meaning of Chinese without a good context. For this sentence, both 能 and 会 sound good to my native ears. Having that skill implies having that capability, doesn't it?

I agree. The normal formula is会+说(语言). Unfortunately this is an area the Duolingo Chinese course often messes up on, partially due to using 会 primarily as a kind of magical "future tense equivalent" wand, which affects the accuracy/authenticity of many sentences. Granted, 会 vs. 能 usage is a very complicated topic, with many grey areas even for native speakers.

会说（语言），能说【流利的】（语言）are both idiomatic. Not sure how to explain since both are technically not wrong and this is something picked up through exposure to the language (read, listen, etc. more)

他住过中国 is also okay, but I think 住过中国的+{place, e.g. 南方} is much more common. The more exact the place is, the better it sounds. I mean, 住中国 (not 住在中国) sounds more like “dwell/lodge in China” rather than “live in China”.

As far as I know, you would never put an adverb indicating place after the verb.

So in thinking about this, I suspect my answer was rejected because it violates the requirement for location to come before action in Mandarin sentences.

My answer “他住过在中国” places the verb "住过" before the location where its taking place 中国