They all sound ridiculous! I wouldn't use any of them. Try to be more casual. "I teach you English, you teach me Chinese. Eh?" ...or similar.

First, since there are two of you, and you're talking about two different languages, you should use "languages". I would write: "I hope we can help each other to recognize the beauty of our respective languages." "Respective" indicates that we each have our own language. It may sound a little formal, though. "Mutual language" means "the same language": if I was talking to another native English speaker, I might say "Isn't our mutual language great?" (meaning English), or "Let's call our mutual friend to come with us." (meaning we each share the same friend). It is not a simple translation of the Chinese concept 互相, so be careful when you're translating that word into English! It's not easy to do. "Mutual help" and "mutual support" are both correct, but probably a little too formal sounding for this kind of situation.