I live just outside Anchorage, but have spent some time in Fairbanks on business trips. Personally, I think it is a great city. I don't think you can compare it to a similar lower 48 area, though I'm not that familiar with back east or the catskills.
Weather can be extreme -60 in the winter, and 90's in the summer. The last few winters across the state have been very mild, Anchorage area didn't drop below 0 the last two winters, where typically we see -30 for a few weeks, and I don't think Fairbanks was much lower then -30.
The city doesn't have sprawl in the conventional sense, but it is spreadout. Fairbanks really feels like a subburb and not a big city, though there are bedroom communities in outlying areas. I don't recall there being many buildings over 3 stories, and most of the time driving around you just see spruce and birch trees. It has a real college feel to the town, and it is a good college as well.
I haven't eaten at many resteraunts, but the ones I've been to have been good. One was called Wolf Run or something like that, they have good food and outstanding desserts.
As far as how far to go for good hunting, depends on what you consider good. Lots of folks take game not that far out of town, but for world class conditions, you'd probably want to charter a plane and fly out a 100 or so miles. You have access to moose, caribou, black bear (to a lesser extent), grizzly bear, sheep and goats as well as small game and fowl. It all comes down to how much time and money you have to get out after game. If you're expecting to drive 20 miles out of town, hike out in the woods and see it crawling with game, you'll be sorely disapointed.