"My second question is, if you had a few days to explore from Nine Lakes Basin, how would you do it?
I was thinking, to see Kaweah Basin, we could set up a camp at the southeasternmost lake in Nine Lakes Basin (I think it's called Lake 11,600) and then drop down into Kaweah Basin the next day with our day packs."
Your most direct route into Kaweah Basin from 9 Lakes Basin is Pyra-Queen Col, just north of Lake 11,682(Tom Harrison Mt Whitney High Country map) in the 9 Lakes Basin. This is a tough class 2-3 pass between Kaweah Queen and Pyramidal Pinnacle(not named) on the Kaweah Peaks Ridge(see Secor, pp 74-75). Once across the Col,the descent is easy down gentle scree and then slabs into the Kaweah Basin, UNTIL you reach a level, gawd awful talus field that goes on for about a mile. It will test your legs, and head. Once past that you will find yourself in what is, IME, the most beautiful place I have ever visited in the Sierra, or just about any place else, for that matter. It is well worth hanging out for a couple of days to explore on day hikes. It would be a bit ambitious, to put it mildly, to try that route as a day hike, and a missed opportunity as well, IMO. Better to break camp in 9 Lakes Basin, go on over, and spend a couple of days. From there you have several ways to exit the Kaweah Basin, depending on how you want to return to Mineral King: 1) Descend the unnamed East Fork of Pickett Creek to the Kern Kaweah River and either head west over Colby Pass, then south over Elizabeth Pass, etc(Dave T knows this section much better than I), or drop into the Kern Canyon(Junction Meadow) and head down to Rattlesnake Creek and up over Franklin Pass; 2) Follow the chain of lakes in the west arm of Kaweah Basin down to the north rim of Kaweah Basin, descend diagonally NW along a seam in slabs to the Kern Kaweah River, and proceed as outlined in option 1, above. The following link will point you to a detailed description of this option that is part of a trip report I posted on the Kaweah Basin last fall(skip to Day 5 if you don't want to bother with the whole report). It should give you a reaonable idea of what is involved(http://www.backpackinglight.com/cgi-bin/backpackinglight/forums/thread_display.html?forum_thread_id=25394); 3) Ascend the west arm of the Kaweah Basin and cross the low pass between Lawson Peak and the southernmost of the 4 Picketts that, along with Pickett Guard Peak, form the west rim of the Kaweah Basin. Descend down into the headwaters of the Kern-Kaweah River over extended scree slopes and pick up the Colby Pass Trail, from whence you have the same options described above. All of the options will take you through some magnifcent country, and will test your stamina and, to some degree, route finding skills. All can be done in 7 days(assuming 2 days from Mineral King to 9 Lakes Basin, allowing a couple of days in Kaweah Basin for exploring.
"It looks to me like (some of) that might be possible in a long day hike if we set up camp at Lion Lake or Glacier Lake, but I have no recon info about whether either of those lakes is hospitable for camping."
There is an excellent campsite between upper and lower Lion Lakes, a nice sandy bench with a perfect tent site sheltered by a huge boulder. Glacier Lake is campable, but not as hospitable as this one. A lot depends on how you want to position yourself for your explorations.