How do you say "will you drink?"
I think that would be क्या तू पियेगा?
Yes. That's correct. You can also go for "तू पियेगा क्या?".
Wow this is so much like Japanese, "Thu Kya Piyega" feels like a casual dialect. And I think if we use this dialect to anyone older than us or someone with a higher status/position/class, I bet it sounds rude.
Very. From what I have read, you should only use तू with close friends, significant others, etc. (though it varies from region to region).