Firstly, Bunny, you're going to love the trip.
We took the trip twice, once this exact trip with GCT, and once a very similar trip with Saga Holidays. Our GCT trip was in early June, and except for Lake Powell, it was pretty chilly. The Saga trip was in early July and it was generally quite warm throughout. For your time frame I would expect warm to hot conditions. Pack for a "summer vacation" but throw in a fleece hoodie, sweater, light rain jacket, whichever best fits your style. You should not need any heavy clothing. Well, maybe a little warmer if you plan to take the float trip on the Snake in Jackson Hole. That is likely to be the coolest time you will experience. Your raft driver will have ponchos for you in case of rain.
We took the Badlands extension and were very pleased. The Badlands themselves are worth the trip, but the additional time in the Black Hills, and the time for touring in Rapid City were also very pleasant.