How would you describe his character?

The character seems arrogant and self assured. As it turned out he had too much arrogance because he doesn't know that 75° below zero is an impossible weather to travel in. If he had waited until the next day when the weather broke and was nicer he may still be alive. So his character is foolhardy but brave.

Is his self-reliance an exercise of irresponsibility and folly?

In the first few paragraphs the character is described as a newcomer and the author warns you that this is his first winter. I believe the character thinks he is being self-reliant and brave it turns out to be folly but because he is new he simply doesn't know better.

Or does he embody an admirable spirit of adventure and enterprise of the sort that tamed the wilderness, settled the West, and established businesses?

If there wasn't crazy people like this guy going out West and trying to figure it out no one would know about the wet foot thing that you have to build a fire right away and no one would know that 50° below zero is the cutoff point where you shouldn't go anywhere someone had to do it first for them to figure out these roles. If it wasn't for these crazy people no one would've ever figured it out.

 

What do you think of the old-timer's advice and the reservations clearly expressed by the dog?

The dog knew that it was not time to travel but he was afraid of being whipped so he went with his owner anyway. After his owner died he followed the trail himself looking for a hot meal and a new owner. The difference between the man and the dog is that the dog knows better. The man is simply ignorant of what this kind of weather can do to you but the dog is smarter.

The author called the dog the man's toil slave and he had never known a kind caress only a whip so I'm not sure why the dog would put up with this crap and not just run away.