Hi,
I cannot help about accomodation, I guess depents on your budget and preferences... But in any case, you should book in advance.
About wheather, it's usually cold, that is 10ºC-15ºC (that's cold for me

Things to see. This year we're celebreting Gaudi's 150th anniversary, so you can visit some of its buildings, The Sagrada Familia Cathedral is a must. There's also a bus (Bus Turistic) that drives you around different touristic places, you leave the bus, visit de monument, and take the next bus.
Here the website of the city council:
http://www.bcn.es/english/ihome.htm
For New Year's Eve, there're a lot of parties organized by discos, restaurants, hotels... You can expend the night long dancing in those places. You buy the ticket and you get all the stuff you need to have fun (drinks, globes, serpentines...)
Hope this helps, if you need more information...