Yes, I think it's an important aspect. Language learners need to be aware of the appropriate ways to address people, express gratitude, make requests, and agree or disagree with someone. So basically, they should know that behaviors and intonation patterns that are appropriate in the language their learning so they won't sound rude or impolite.

Always! and vice versa

Absolutely. A language is constructed within a society. Their habits and customs helps to build it directly everyday. In the other hand, to learn a language is to get a little in touch with other people's way of living. For instance, getting in contact with the chinese culture helped me a lot in the learning of chinese.

yes,of course as this encourages you to learn more about the history of the laguage and how it last for all this time.

Yes, it does. Because a Language is part of a culture.