To your original question of whether you can make a 4,300-mile trip in 9-10 days; you might be able to, but I couldn't. Doing the math and using a reasonable 50 mph average for the trip, that's 86 hours of driving. That means just over 8-1/2 hours of driving time per day for 10 days. So, by the time you factor in rest stops, meals, etc. you would probably have 12 hours each day "on-the-road."
If it were me, I would fly to Alaska and extend my visit there accordingly. You indicate that you would like to drive through Canada (presumably to see the sights) but you won't have much time to take other than a fleeting glance out the window at the countryside.
Just my two cents worth.