Hi!
I lived in the Twin Cities for 6 years while in grad school at the U of M, so I know the area well enough. First of all, I'd recommend that you consider Saint Paul instead of Minneapolis. Housing is cheaper, crime rates are lower, and it generally has more of a "neighborhood" feel to it. I've lived in Minneapolis (Marcy Holmes neighborhood, close to the U of M) and in Saint Paul (Dale and University area) and I definitely liked Saint Paul more. You might consider the Summit Hill/Crocus Hill neighborhood, or somewhere near Grand Ave or Selby Ave. Also, the new Central Corridor light rail line was just completed - it runs between downtown Saint Paul and downtown Minneapolis, so living anywhere along that line gets you to the U of M quickly.
To respond to some of your specific questions and preferences:
Everywhere is LGBT-friendly! Everywhere is bikeable! The Twin Cities are awesome.
Uptown is very young, busy, and trendy (read: expensive). Rent will be higher there and it will be hard to find a place that meets all of your housing criteria. You'll pay a premium if you want to park two cars. It is very walkable and has a lot going on entertainment-wise. The Northeast area is a little less trendy than Uptown, but still has a lot going on. Housing might be easier to find here, but it's still way more expensive than Saint Paul. If I had to pick between those two, I would pick Northeast. Avoid anywhere right next to the U of M - those neighborhoods are dominated by drunken undergrads.
If you're not used to winter in general, be warned: it gets COLD. You can experience this level of cold by visiting your local grocery store and asking them if you can hang out in their walk-in freezer for half an hour. Luckily, as MMM has pointed out, the solution to dealing with COLD is "clothes". "Down" and "wool" are your new best friends! It gets dark early and winter can be depressing if you decide that it's "too cold to go outside" (a condition which does not exist, in my mind). Put on those layers and get out there! Discover activities like pond hockey, cross country skiing, fat-biking, and maybe ice fishing?
Give the winter biking thing a try! I did it for several years and continue to do it now (in north-central Wisconsin with the same climate). There are MANY hardcore bikers in the Twin Cities, and they are happy to help and give advice. Really, if you stop by the side of the road with a bike, someone will inevitably stop and ask you if you need anything. I think it's the #2 biking city in the US (behind Portland), which is kind of amazing considering that it's one of the coldest major cities in North America. Again, the secret is "clothes". Ski goggles, good gloves, snow boots, and LAYERS are essential. Traffic sucks in the Twin Cities, but in the worst winter weather, biking was actually the FASTEST option for my seven-mile commute. Start doing it in the summer or fall when the weather is nice and just don't stop.
Heating costs will completely depend on how well your building is sealed and insulated. I had a 130-year-old house in Saint Paul and when we moved in, we were paying something like $250-$300 per month in the coldest winter months to heat it. After a vigorous application of insulation and sealing, that came down to $150. You won't have this option if you're renting, so make sure you ask to see heating bills from the past two years for the specific place you're considering - I think a landlord is legally required to provide these. A/C costs will be much lower - find a place with some good, big shade trees on the south side and you'll use it sparingly.
As far as housing costs go, keep in mind that you're comparing the Twin Cities to SF, which has some of the highest costs in the nation. $3000/month may seem cheap to you, but you can probably find most of what you're looking for and spend less than $1500 if you look in the right place (again, the right place is "Saint Paul"). Check Craigslist for rentals, the postings are usually legit.
They do sell windshield washer fluid that's good down to something like -40F, and you might want to add some additional (undiluted) antifreeze to the engine coolant to give it a bit more freezing protection. Lower viscosity engine oil can help with your car's health - check the owner's manual (or do some online research) for recommendations. If you find that you are engaging daily in the insane habit of starting your car to let it warm up before you drive it, invest in a block heater instead. I would STRONGLY recommend snow tires (I agree with MMM's points in his article on snow tires). I also recommend the bike equivalent of snow tires: studded tires.
Hope this helps!
