Can you add at least one day?
Two more days would be even better.
I agree with glc's suggested route. It's definitely the shortest/quickest way to get to Seattle. However, I think you're pushing it doing it with two small children and two old dogs...all who will need time out of the car to stretch their legs, burn off energy, etc. If you have unhappy kids and dogs, it's going to make for a miserable trip.
Having traveled with kids and dogs, I can tell you that you will want to stop at least every two hours. Look for nicer rest stops or go into a town and look for a park. Let the kids run a play for 15-20 minutes and get the wiggles out. Let the dogs do their business and sniff around.
The trip will be far better for all of you if you do this. If not, you might get to Seattle pretty stressed out.
I didn't check out all the suggested overnight stops but I can tell you that Green River to Boise, and Boise to Seattle, are both about 500 miles and, once you add in time for those stops, will make some real long driving days.
Personally, I think you need to plan for no more than about 400 miles per day to get those needed breaks from the car in and for you all not to be too exhausted and tense by the end of each day.