make sure you dress appropriately, you can actually get sick from over-exerting yourself in the cold (think about it. you sweat to keep cool but your out in the freezing cold... so extended periods of intense activity I wouldn't suggest) Cold air is harsh on your lungs and makes it difficult to breath. I train in the cold in winter and it helps me cardiovascularly... just make sure you know your limits.
as for running a mile in 8 minutes... who cares if it took you nearly 13 minutes? At least you tried and I'm sure you're planning on trying again since you posted these questions here. I can gurantee you are already doing better than 50% of the people I know
Just some advice:
When you are running try not to think about it as 'oh I have to run this in this amount of time', it is all based on your heart rate. So the way that I was shown to teach others to get to their target heart rate was just to run until you can barely say one sentence without needing to take one breath. Do this for about 20-30 minutes and you'll notice great improvements. So yes, this means if you walking puts you into the 'zone' then you should just walk. You won't really get anywhere if you just run hard at the beginning and get winded really fast... there's nothing to be embarassed about.