As someone who's a native speaker but not a teacher, I had to think on this one a bit.
I'd say that 'would you mind' is usually used when you are making a polite request for someone else to do something or asking if a hypothetical situation would bother someone.
"Would you mind washing the dishes?"
"Would you mind if I listened to music?"
I feel like I most frequently hear "do you mind" in an impolite context, like if someone is chewing loudly you say "do you mind?!" to imply that you want them to stop. But it can also be used to ask someone if something bothers them that is currently happening in that moment, or that happens habitually.
"Do you mind the smell?"
"Do you mind when she listens to music?"
I don't think I've ever heard "will you mind," it sounds kinda weird.
Bear in mind that this isn't based on confident grammatical facts or anything, just my perspective as a native speaker. Hope this helps!