Mid October is a great time to visit so you picked a great time. There should be quite a bit going on in the mountains, fall foliage, Oktoberfest type of stuff that's quite a bit of a tourist draw. You should be able to do outdoorsy stuff like you mentioned as well. I actually would say Red Rocks is a must do, it's open year round to go to the park and amphitheater but yeah by October they don't have much for concerts.
As far as the hotel, I'm not certain on what's around there (I never stay so I tend to not pay attention lol) but if it's closer to the football stadium on the other side of I25, I'd advise not to walk. I think there is some way to cross by foot which involves going by the railroad tracks but seriously there is no need to do it. There should be no issue parking at the arena or if you don't want to drive at least take the light rail. There is a stop right at the Pepsi Center and there are a couple around the football stadium that would be just one stop away if you got on there.
There is only really Brooklyn's by the Pepsi Center. It's ok but does get a lot of traffic. Going a few blocks into downtown gives you infinitely more options, check out the area of 14th and Larimer as there are many restaurants there and many good options depending on what you like. The Pepsi Center is situated between the highway, amusement park and large college campus so it's not right IN downtown though it's just a few blocks away so it depends on what you envision as right by. Walking distance sure but not exactly "on the way" to the game.
Hope that helps...