Welcome to RTA!
Having crossed the country many times on both the routes you mentioned, I'd go with Route #1. It's at lower elevations, there are less mountain passes to really worry about, and in many ways, it's a far prettier route.
Your year-old son may not be the worry you expect. I have grandkids and both of my own kids worried about how their children would travel. It turned out to be much easier than expected. And honestly, diapers are easier to deal with than a potty-training-child!
As for your dog, have you talked to a veterinarian? The vet may have some solutions for you. Most dogs are great travelers, but there are always those that aren't.
Once you get the route ironed out, you'll want to space your overnight stops about 500-550 miles apart. That will give you a solid 10 hours a day on the road, with time to stop to change a diaper, check out a scenic viewpoint, etc. Because you are traveling with a dog, you may want to reserve your overnights, as well, to get the pet-friendly hotels.
Are you traveling with a Pack-n-Play? Both of my kids have taken theirs with them, as hotels may or may not have a crib available for the baby.
Believe me, you will get experience and will feel more knowledgeable as you travel more!
Donna