Here is what I would do.
1. Watch how the snow year shapes up. In a big snow year, I would hike the JMT beginning in early to mid August. In a regular snow year, I might start around July 20. In a low snow year I might start about July 10. Right now it is too soon to tell how the snow year will shape up. All this early snow/rain is a big anomaly. My goal is to avoid a lot of postholing, or drowning in a creek crossing, or being eaten alive by mosquitoes. So I like to wait until the snow has melted, the creeks have calmed down, and the big moskie bloom has subsided.
2. I would get a northbound permit starting from Cottonwood Pass or Cottonwood Lakes. I would camp at Crabtree and dayhike the summit of Whitney, in order to get the "full trail experience".
3. The permit process for Yosemite got a major revamp this year so I couldn't tell you anything about that. I hope it is an improvement.
4. The permit process for Inyo has been much the same for many years, and I have always gotten the permit I wanted. I guess I have been very lucky. I typically get my permit 6 months in advance, which would usually be late February. But if conditions change I may drop it and pick up a different permit for a different date or trailhead.
Disclaimer:
I hiked the JMT nobo in 1996, starting at Camp Whitsett in Johnsondale.
I hiked the JMT sobo in 1999, starting at Leavitt Meadow near Sonora Pass. Since then I have hiked segments every year, always starting from the east side, so I have gotten all my permits from Inyo.
