Don't get me started, I'll talk forever.....my kids get upset with me because when hiking, it may take me and extra hour or two to reach the car.
So, I am happy to answer your questions.
A couple observations: I used to take Cottonwood Pass when heading down to the Kern, then one trip, I decided on Trail Pass. The obvious reason, no trail quota, none!!!! Second, it's about 1 and 1/2 miles shorter via Trail Pass to the Golden Trout junction. The pass is lower and finally, it's only 2 miles to the pass from the trailhead. Trail Pass is fairly easy to hike coming out of Horseshoe, however, I would classify the back side (on the return, moderate).
My first night camping is at Groundhog Meadow (about 12 miles).
Mulkey is sand and open, no trees....your first water is the South Fork of the Kern (about 6 miles in). From there you have water through Tunnel Meadow.
Your question about the trail being "overgrown and not maintained." I was referring to the section of the trail that occurs after the Natural Bridge where it starts it decent into the Kern River. When I say, "overgrown", it's not really that bad.....I just hate it.
Until you reach the decent the GT trail is very well maintained, travelled and visible......a gentle slope on nice ground, through trees with some nice side streams.
Once you reach the bottom, I also hate this, the trail heads down the Kern for about a mile (sand) before you come to the Kern River Bridge.
The trail heading up the Kern is very gentle, heavily used. I will say this, the trail has not been cleared of fallen trees and such last year, and so far this year. Not a problem, easy to get around them.
Please don't get me wrong, the Kern is a very beautiful place where the trail rarely meanders far from the river.
As for my comment about the trail from Wallace to Lower Rock Creek.....take the comment with a grain of salt. I think the reason that I like that section is that, after several days in the Kern, it's time to hike out. So this section is close to the end of the hike and therefore my favorite.
Once at lower Rock, I usually head up Rock Creek and camp at the junction of trail heading up to NAP (most times though, I take Army....why climb so high). And, I always take the South Fork Trail out...saves miles and time.
Also consider the Tyndall Creek trail that breaks off the trail to Milestone. It's about 1.2 miles to the junction of the HST, out of Junction Meadow. There, it's about a mile to the crossing of Tyndall creek and another mile to where the trail leaves the Milestone trail and heads to Tyndall Creek.
When I take this route, I usually hike 12 to 13 miles the first 2 days..... First night Groundhog Meadow (Little Whitney is only 2 miles farther). Second day at the junction of Rattle Snake and Kern River trail....third day at Junction Meadow (when I don't stop at the hot springs). It's 17.5 miles from the Kern Bridge to Junction Meadows. By the way, your calculations are dead on.
An alternative.....head up Wallace Creek...take a couple hours to do this, head back and camp at Wright Lakes....(Wallace and Wales in only a few miles from the junction of the JMT....easy hiking). From Wright lakes and then head over Rockwell Pass to Shepherd Pass.
Hope I am not too confusing....
paul