Dogs can do lots in Alaska. Ours have travelled with us throughout the state. There are few limits on where they are permitted to be, other than in Denali. There are many, many dog owners in Alaska so dogs are accommodated and tolerated in many places and you'll not find any breed restrictions.
You can take your dog into Denali Park in your vehicle and you can walk your dog on the road in the park but you cannot walk your dog on the trails or anywhere other than the roads. Your dog MUST always be leashed in Denali. Your safety is one reason - dogs tend to bring bears back to you. As well, they don't want your dog harassing the wildlife in the park, and don't want the wolves to be affected by presence of a foreign canine in the area. We did the day bus tours into the park, each of us going a different day so the other could stay at the RV with the dogs.
We're not big into commercial tours, so had no worries there. We tend to spend a lot of time outdoors, hiking remote areas, exploring back roads, etc. so our dogs get a lot of exercise and fun time. Our dogs know bears so we aren't concerned about their behavior there. They keep the bears away, not bring them back.
A bonus is the dogs alert us when there is wildlife hanging around the RV or around us when we're hiking.
We did head off on an extended trip with a six-month old puppy who we had adopted a few weeks beforehand. She did exceptionally well, learned from the older dog what worked and didn't, and we kept a fairly steady canine routine on the road. After that trip, she was a well-seasoned traveller and exceptionally well-behaved while camping. It was her normal.