A fat tire bike is a type of mountain bike that runs extremely fat (hence the name) tires that are run at a super low pressure. They tend to float in certain types of conditions and terrain, e.g,. sand and snow.
Here's an example.
A fat tire bike, in my opinion, would not be the best choice for what you've described. They're not typically particularly light and they'd bog you down on pavement.
I wouldn't necessarily rule out steel just because of weight. For one thing, not all steel bikes weigh a ton, but keep in mind, too, that weight has as much to do with the frame itself as it does with how that frame is built up. So, let's say the Tricross frame, with for, weighs in at 4 pounds (which is one of the reported weights I saw for the bike) and the Cross Check comes in at 7 lbs with frame and fork. Assume you build up the bikes the same way, we're talking about a 3-pound difference. I'm not sure that's enough to be a dealbreaker--or at least it wouldn't be for me. The ride quality, however, for steel can be really worth the weight penalty, especially if you're riding the bike offroad. Just something to keep in mind......