During the summer Suomenlinna / Sveaborg and/or the sea cruises would be nice, but not in December. Stroll around the town if it's your first time, check church / museum / galleries etc. depending on your interests. All the xmas buzz is at full speed in December, there are some concerts (usually in churches) that your day might match.

Nope, usually the cold front hits in January when it can go down to -20..-25 C for a week or two.

December weather can be a hit or miss, hopefully there's snow because it makes everything so much better. Anyway, unlike some other countries we keep the outside and inside temperatures separate

Depends on your budget but Klaus K and Hotel Glo are good value for their style (design) and location.

The upscale hotels in Helsinki are pretty decently priced, even at Hotel Kämp you can book a room for December for 216 €.