Hello and welcome to the RTA forums !

Another viable option to your route above on I 82/84 through SLC and so on, would be I 90 to Buffalo WY, I 25 through Denver to Raton and across to Amarillo. The alternative route is 160 miles longer, but only an extra hour travelling time according to my mapping program and could be considered as a back up if weather and/or road conditions are not favourable with plan 'A'. You will only get a true picture of this when the time comes close to leaving.

It's been a long time since I travelled with a dog but we used to start the day out with a good exercise so she would then settle for a good while and then get a few miles under our belt before stopping for short and regular breaks. You have 4 days where you will be in the car for 9/10 hours a day so you will need to find areas close to your route where you can all have some 'R&R' and this great thread should provide you with a few ideas.