Welcome to RTA!
If you plan on your days being about 500 to 550 miles, that should be plenty of mileage to feel like you're getting somewhere, and give you relaxing time out of the car to walk the dog, etc. You can use an electronic mapping program for getting distances, etc., but beware of using their fantasy travel times. Always add 20% time to those, or just divide your distance by 55 miles, to get your average travel time (550/55=10 hours, for instance!).
1) Well, of course, beauty is in the eye of the beholder. But if this trip were my husband's and mine, we would say "I-70". Between Denver and the end of 70, in Utah, is some of the most gorgeous scenery that any interstate in the US gives us: the mountains just west of Denver, and the San Rafael Swell area between Green River and Salina, UT. That latter 108 miles has quite a few scenic pull=outs. Humans beware, though, that most of them have dog-walk areas but no human potties. :-) You'll have to take another route to get into San Francisco. I'd recommend doing US-50 across NV if you can, or going up I-15 to catch I-80 if the weather patterns in Nevada don't look good. US-50 is a little slower, "the loneliest road", but more direct. It will take you to I-80 anyway, up and over Donner Pass, and west into San Francisco.
I-40 is scenic in its own way, but is probably a bit more mileage for you. Going up to 80 would be another option, but it takes you through the traffic of Chicago (yuck) and is not nearly as beautiful anyway.
2) Pet friendly lodging usually includes Motel 6 and La Quinta Inn. Others are hit-and-miss. The 120 lbs might be another limitation. You will have to figure out how you are going to handle meals out, however, as most motels (M-6 and LQI included) don't usually want you to leave Fido unattended in the room.
Most of the interstate rest areas have dog-walk areas, and most states are even courteous enough to provide plastic bags for you to clean up your pet's mess. There are city and town parks almost everywhere. Hays, KS had a lovely city park not far from the interstate, and if memory serves, had a nice area for dogs, too. But that's just one example.
Donna