There is only about 10 miles difference between the route I suggested and taking I-90 across the northern plains and over the Bitterroot Mountains, and in fact the two are so close, that my route is shorter to Walla Walla, but the I-90 route is shorter to Richland. You should also know that if elevation is what's most important to you, then the I-90 route stays lower (around 6,100 feet maximum) but the climb is steeper than on the I-72/I-80/I-84 route (8,500 feet). There really is not a lot to choose from between the two routes, so have a look at them both and pick the one that appeals to you.

As for how long the moving company will take, no one will know better than they will. Ask them now, and ask your driver when he picks up your stuff if you can.

AZBuck