First, the last two questions. In LA, I recommend Bel Air Camera in Westwood. www.belaircamera.com
. About 20 minutes north of LAX unless you hit it in the afternoon rush hour. There is one camera store - Sammy's - in Culver City which would be about 10 minutes from LAX, but Bel Air Camera is a more extensive shop.
Lenses. If you could replace the 24-70 with the 24-105 you won't regret it for this trip. You can use a tripod in Upper Antelope, but in Lower it can be very difficult. The IS of the 24-105 is very welcome in that instance. Otherwise, you are on the right track.
Parks. I think you can already see from the replies there is no right answer. We all have our favorites. Mine are Yosemite, Page area (but not only for Grand Canyon), and the Grand Tetons. In my view, of all of these, Yosemite is the treasure. Six hours from LAX by car. I can't imagine missing it.
You can do the slot canyons and Horseshoe in Page all in one day. At that time of year, I think you should start with sunrise at Horseshoe. Only bring your widest lens for that trek - you will not need the others and won't want them on your back on the walk back up to the parking lot. It will take you about 20 minutes of a walk on an open but sandy and incline path each way from the parking lot. It is just 5 minutes outside of Page. Then absolutely hit Upper Antelope Canyon between 11 and 1 - which is about 10 minutes from Page - for any chance of the sun shafts. The canyons are two experiences - one with direct light and one without. From Upper, go across the road to Lower. You can spend more time in Lower if you pay an additional fee. (When in Lower - pay attention to the rope ladders when framing your shots - they lend an interesting perspective.)
The other replies have covered a bunch of other great ideas so I won't repeat them. I would say, however, if you can find a way - long drive, plane - to get over to Jackson, Wyoming from the Southern Utah area, you will not regret it. The Tetons are stunning, the area is unbelievably gorgeous (any time of year), and Yellowstone is unique in this world.