The former. Deer, furs, whales, and silver tend to be found in arctic areas and nowhere else.
Yes, that works too, but for a city with extremely low production and food (which sounds like what you're describing), even a relatively cheap build like a theatre can take a long, long time. To get the first border pop either a missionary or building culture (once you have music) are more efficient.
As for chopping--keep in mind that in an arctic area you want to be very picky about chopping trees, especially those on tundra tiles. If the tundra tile is immediately adjacent to a water source (lake or river), chop away; you can farm the tundra for 2

pre-Biology, 3

post, which ain't bad. A tundra tile with no fresh water source, however, cannot be chain irrigated or improved with much of anything at all. If, however, there is a forest on the tile, at least you can build either a forest preserve or a lumbermill there.