As for itinerary, in my book 8 running days should be enough going to the Sturgeon through Stouffer: e.g. camping on Wolf, Chiniguchi lake, Stouffer (3 relaxed days that might be done in 2), Sturgeon at the Pilgrim Triangle, Upper Goose Falls, Kelly's or Gawasi lake, Kukagami or Donald. It can be done in 7 days as well, depending on how fit and enthusiastic your group is. Going north of Stouffer - through Backdoor to Kettle falls - will require about 1 to 1.5 more days. Portages on the backdoor route are short, but some of them are pretty tough. Same goes for the Sturgeon portages upstream of the Pilgrim Triangle.
My calculations are based on solo trips, double-carry portages, and I'm no athlete.
Parsons and Halleck routes include lots of walking, mostly on easy roads, but still... I like these routes, but for a first taste of the area my choice would be the river at least from the Stouffer to Kellys' portage. When I say Kelly's portage I mean the Kelly-Gawasi lakes route (never did the 3.4km portage). Campsites in this part tend to be very basic and beware of the first portage out of the Sturgeon: the split (where the 3.4km goes straight and Kelly's lake portage goes left) takes place in a grassy area with no visible points of reference and the trail to the left could be almost invisible in high grass.
All my descriptions are based on Ottertooth and Hap Wilson maps, which in my opinion are the only reliable maps of this area.
Permits are required only for the nights you camp on the Sturgeon.
In my limited experience, the Sturgeon in July usually had no serious water level issues. At its worst you might be getting out of the boat here and there on gravel shallows. Sturgeon levels measurements are available at
https://wateroffice.ec.gc.ca/report/rea ... tn=02DC012 . I comfortably ran the river at discharge as low as 4.50m though it did require some easy wading.