I have a multi-language support website.
My question: can I show/hide some fields based on the user's language choice with Rules module?
The Conditional Fields module might be of help to you.
Refer to the video about Field formatter conditions for a demo of what is pretty close to what you're looking for. Especially starting from around 4:50 in the video, you can see the Rules action "Hide a field".
So with that, your question is reduced to something like "How should my the event and optional conditions for the Rules module look like, related to the user's language choice (to apply the "Hide a field" action)?". The answer to that question could be another question of course ...
Note: the article about Introducing Field formatter conditions for Drupal provides some more background reading about this video also.