Driving the Alaska Highway is a great experience, however it is a two lane road, some of which is over tundra. There is a lot to see along the highway. I would advise you ride on the top half of you gas tank. Gas is available but there is not a station on every corner. Drive to the conditions. Slow down over frost heaves and through construction areas. There is a very short season for highway construction and the hard winters are hard on the roads so there are always repairs being made.
Traveling with pets is not a problem. Many folks do it. You must have current rabies certificates and although not required a health certificate can be useful if your pet has any health conditions and you need to visit a vet. We took our dog without any issues.
Now to the firearm question. Unless you have a specific reason for taking a firearm, leave it home. Handguns are not allowed in Canada and shotguns, I believe, need special permits. On my four trips to Alaska I have yet to feel a need for a gun of any type either in Alaska or Canada.
You might want to get a copy of the Milepost which is a mile by mile guide to the Alaska Highway with information on locations for gas, food, scenic areas, and camping.
We are hoping to return to Alaska again this summer ourselves.