BLUF (Bottom Line Up Front) - Z not a family car.
I would recommend you trailer your Z, out of consideration for the dogs, baby stuff, elbow room and protecting your car.
Being single with no kids and no pets, I had the pleaseure of driving my Z cross country with total freedom. I drove my Z from Fort Lewis WA, to Barstow CA, through Dallas TX to Richmond VA to see my family. Then drove to Pittsburg PA before coming over here to North Dakota. Cramped as Hell, but a trip I would definately take again (racing BMW's, cruising the Oregon mountain passes, gently rolling Virginia hills...amazing!)
If you absolutely must take the Z on the road, obviously wear and tear is inevidible, but mine survived the 7 day trip rather well. Just take it easy when you get to NM/AZ. Upgrade the cooling system if you can...otherwise the desert heat will keep you at 230-245* even on cruise control.