Helsinki is great. I lived there for 6 months while I did Finnish national service – hardcore (I'm half Finnish so may be slightly biased).
Everywhere in Europe is expensive since the pound plummeted but I've never found it any different from drinking in London. A lot of Finns drink at home beforehand to lighten the burden…
The Finns have a reputation for being quiet but after a few beers they loosen up. Plus, once they find out you're English speaking they'll want to practise on you!
It's probably worth getting a 3 day travel ticket that can be used on the trams, buses and underground.
As for cold and gloom, they've had their first snow already last week but it's unlikely to settle.