If you really really want to see the Grand Canyon, that's something you could fit into a two week trip. You could get there in a pretty comfortable 5 days each way, being on the road for 6-8 hours a day, but still leaving time to make a couple nice stops each day, and have plenty of time for smaller stops for the dog to run around. It could even be done in 4 days each way, but I think that would require a pace faster than you'd enjoy.

That would be about 10 days of driving round trip, leaving you 2 days for the Grand Canyon, and a couple extra days at stops where you see fit. Keep in mind that New Orleans and Austin/SA are not on the direct path, and that would be another full day of driving between NC and Grand Canyon.

San Diego would be tough. That's another full day driving past the Grand Canyon, so while it could be done, I don't think you'd want to do that kind of pace.

And you will have to factor in weather. Really, anywhere on your route you'll have the chance of seeing snow or ice. Remember, the Grand Canyon is up at about 7,000 feet elevation and sees snow, but even I-10 across Texas and New Mexico can see winter storms.