The two bars at the W are upscale places. I wouldn't go in wearing T-shirts or even polos (for the guys I mean). Drinks are pretty reasonable in the main level "Living Room" bar. They get spendy IMO up at Prohibition but I think you would find it worth it to get up there for at least one drink. Very cool setup and layout. Almost feels like you're in the board game Clue sometimes.
Why do you need to drive to the Dome? Just wondering. Parking is expensive. Could you take the train?
Anyway, I would stay along Nicollet Mall. You can walk to many bars from there and hop the train to the game if need be. Kind of in the middle if the action there -- the closer to 5th Street the better by the way. Then you can access the Warehouse District area (1st Ave North) by foot if not too cold).
Country-type bars downtown? I'm no expert.
Cowboy Jack's (too me a little heavy on the kitsch from my one visit to the Bloomington location) just opened up a megalocation downtown with a rooftop/fire whjich could still be in play in October with the fire pit going.
Also there is Whisky Park, which I know absolutely nothing about. So you might wanna read up on some reviews.
If you want a slightly older crowd consider popping in at Lyon's Pub on 6th.
Not sure about places where country music can be heard live downtown. Maybe Whisky Park? God knows the place is big enough.