You are asking just that, among the set of options "all the languages" which one do you speak. Same if you ask about a car, when you ask which car do you drive? the set of options is the available cars. If you ask what is on your mind, then there is a difference. But to me, which language do you speak? sounds perfect.
I agree. While I absolutely understand vitaoma's point, because there's a possible context he's correct, "which" could also be correct. Maybe a person grew up in a country with multiple official languages, and is being asked which one they think is their native language. So I absolutely think "which..." should be accepted.
Yes, there is a relation between род and родной. No, sexual gender is пол in Russian. These are not related. I believe you are confusing "gender" with "sexual gender". Gender in English comes from "gendre" French, which comes from "genus" Latin meaning "sort" or "kind." Род has precisely this meaning, not the sexual one.
Google translate is utterly unreliable. The masculine nominative form of the adjective is "тёплый". "Fish" is a feminine noun so the correct nominative form of the adjective is "тёплая", but for some reason on "warm fish" it gives you the answer in a different case. Just now it showed me "тёплой рыбы" which is genitive case, the nominative is "тёплая рыба".
Edit: for checking different forms of a word, this site is immeasurably better than GT. The whole site is in Russian so it may take a bit to learn your way around but it's well worth it. http://www.morfologija.ru/%D1%81%D0%BB%D0%BE%D0%B2%D0%BE%D1%84%D0%BE%D1%80%D0%BC%D0%B0/%D1%82%D1%91%D0%BF%D0%BB%D1%8B%D0%B9