AColumbia (Missouri): Cat friendly hotel
I stayed at the Ramada by Wyndham Columbia 901 Conley Rd, Columbia, MO 65201(right off I-70) it was clean, comfortable, convenient and they allowed my cats as I passed through town.
AOmaha to Phoenix: Drive from Omaha NE to Phoenix AZ via Colorado Springs
Depending on how you're going into Phx you'll have some mountain roads (Flagstaff vs splitting off at Holbrook and going to the east side via hwy 87) but otherwise the road is a pretty easy drive through CO into NM. Time of year can affect the Raton Pass but shouldn't...