Is it possible? Yes of course it is, but would I do it? No. More power to you if you really want to, but like a few others have said, a single speed would be a more casual way to go for the down hills. If you are going to do it on a fixie though, please use brakes! With the weight of your gear and the hills you will experience, you will either have a heck of a hard time keeping in control (or not exploding your knees) without brakes. I have a fixie with brakes, and yeah I am not cool for having them, but I can stop before I hit things.

As for your gearing dilemma, have you thought about running a compact double with a Surly dingle cog in the back? This would give you a much wider range of gearing for long climbs and such. Let us know how it all works out.