I do mostly day trips or sleep in my car, but when I camped at Dorst it was quiet and a little more secluded than lodgepole. Lodgepole is larger, has more amenities, very developed camp sites and is in a cool canyon, but it's also usually busier and has more day visitors since it's a popular trailhead. Lodgepole can get colder at night too if that's a factor, but August won't be bad. Hmmm... Better swimming holes at Dorst, but you'll probably be further from water than at lodgepole.
You can't really go wrong either way though, they're close to each other and both great in their own way. Hope you're planning on bringing your climbing gear 'cause you'll be minutes from chimney rocks, big baldy, little baldy, the watchtower, moro rock, buck rock and a few others!
Oh and I've never been in the group campgrounds so I can't help you there.
Enjoy, it's a great area!
Edit: This is just off the top of my head so I might be wrong!
I spent my last night in Dorst in the early 1950's, but about all I remember is having a great time with my family.
I go to Lodgepole relatively frequently on day trips, and there are many wonderful hikes ranging from short and easy, such as Tokopah Falls, to rather strenuous (if you're coming up for the day, as I usually do) like Alta Peak (out of Wolverton). I wouldn't overlook the hike to Mitchell Peak, off the General's Highway. Although the distance is modest, and the elevation gain moderate, the view is outstanding.
I would say the scenery at Lodgepole beats that at Dorst by a lot, but Dorst is quieter.
Incidentally, I, too, greatly enjoy Cedar Grove as an alternative to Yosemite Valley. If nothing else, the climbing is much more adventurous, but it's quite a trip from there to the main attractions of Sequoia.
Lodgepole. Close to trails, amenities and many of the sights with the exception of those down by Moro Rock and Crescent Meadow. Nut job had a good suggestion with Cedar Grove, but that's in Kings Canyon. It is rad but protected by a long, very twisty drive that will punish kids' stomachs. When mine get a little older, we're there, but not yet.
As far a top-ropes, crystal wall at chimney rocks has some. Also, it's really fun to belay off the west side of the railing on top of moro rock, but it's a lot of exposure for little kids. There are random bolts on top of boulders in a few areas too if you look hard enough. Other than that, find a rock, look for bolts or build an anchor, and let them have at it!