Yes to #1. I took a train there on a whim from Kraków. It was very pretty to see the snow covered terrain on the way in but otherwise all my advice is for inside Pest.
Yes. Drink a lot. But before Christmas because lots of cool bars close then until it warms up.
Also visit Christmas markets.
Also go to rave parties in the Turkish baths. It's the closest to an orgy I've ever been to but in a good way: https://spartybooking.com