We've driven across the country nearly a dozen times - in all seasons; it can be done, so ask away 😊 You can avoid snow in most of the country by heading south, then cutting across I-40, although you'll still likely have to maneuver through the local snow in New York and thereabouts. You may get snow around Flagstaff, so if that's in the forecast you could cut even further south and use I-10. Pretty much guaranteed no snow on that route, and it can be interesting (White Sands NM, Tucson)
We're about to drive from KY to Seattle this Dec/Jan with our cat (and did a similar drive from KY - Seattle - AZ - KY two winters ago without the cat), so have been doing lots of research into pet friendly lodging. Our usual interstate hotel charges crazy pet fees (and we obviously can't camp like we often do), but Motel 6, La Quinta, Red Roof Inn and Econolodge have either no fees, or they're only $5 - $10/day rather than $75+. We've also learned that "pet friendly" ALWAYS means dog-friendly but only occasionally means "cat friendly," although that is neither here nor there.
Other road-trip tips:
* always keep emergency supplies, just in case (and particularly in winter). We always carry 1-2 gallons of water, in addition to some non-perishable food items. In the winter, this also includes flashlights, and warm clothes/blankets.
* We find that eating take-out gets old after a couple of meals, so keep a small cooler with the kitchen basics (a few condiments, bread, whatever you consider "essential") and then a plate/fork/knife/spoon/cup. That way we can always swing into a supermarket or farm-stall to get fresh fruits, cheeses or meats to complete the meal.
* allow for leisure time on the road and unexpected detours so that you get to experience the "road trip" part of the trip. You can always decide to keep driving and get there faster, but if you come across something really interesting, it's a bit obnoxious to have to think "I really want to see this, but there's no time!" because when else are you going to be driving through that way?
* take frequent breaks. We stop at EVERY state crossing as a rule (which on the East Coast is every couple of hours) whether we feel like we need a break or not. Stand up, stretch, walk around, let your eyes focus someone other than the road .... always useful.
What a great experience you have ahead of you!