I went over Christmas a few years back and yes it's bloody freezing, wrap up warm. I really liked it, probably not the prettiest place (though some nice architecture if you wander around) but it has such a cool atmosphere. Very chilled out, nice pace and easy to get around.
Personally I wouldn't worry too much about Tivoli Gardens which is their big tourist spot. Very busy and just not worth the money. We stayed near Vesterbro which was a good spot (seen as the hipster area), loads of nice bars, cafes and restaurants. If you stay out of the city centre you can definitely get reasonably priced beer. In the centre you can pay up to £8/9 a pint.
Christiania is well worth a visit. A freetown on the edge of the city which was taken over by hippies in the 70's. Old army barracks and they basically moved in. It sways between being under state law and them running it themselves so people are openly selling drugs from little huts. And the walk round the lake is nice. Just a really interesting place.
Reffen Street Food market is great too, about 20-odd street food vendors in a huge warehouse, DJs, few nice bars etc.