Thanks for visiting the site, but I'd like to make a few comments here for general consumption:
#1- How reliable is the system for requesting permits? If you peruse NWHikers.net on doing the WT, you will see that most people got what they asked for (2011) with little changes. Also, many people postponed their trip due to heavy snowpack and came later in the season, they they fared well in establishing a new itinerary. I have never had permits in advance, I just walk in the day of the trip. Granted, I don't have long itineraries but I don't think I've ever had to change my planned campsites, even in summer. The tough ones will be Summerland, Eagles Roost, Indian Bar and perhaps Berkeley Park due to their proximity to roads and reachable by a single day trip.
#2- Any particular place that is better or worse to start out? After looking over the map, it seems like an ok plan to start at Mowich and have food drops at Longmire and White River. You only have a few choices, so starting at Mowich is as good as any. That way you can avail yourself of the services at Sunrise and Longmire on your trip. I usually start at Longmire, and occasionally at Sunrise as these have major parking areas and facilities.
#3- Campsites to try and stay at/avoid? I know my own stomping grounds (NCNP) certainly have campsites that are less than desirable, hoping to avoid those on this walk. Also, is the spray park detour worth considering? First I'll say that in good weather I would do the Spray Park route...it's not really a detour and is actually listed on the NP website as the "alternate route." Great views. If it's overcast and/or rainy, then the Ipsut Pass route is probably better. Just be aware that the climb from above Ipsut Cr campground to Mystic Pass (going cw) can be a bear. Ipsut Pass also has pesky overhanging brush and hidden rocks so expect to get soaked if it's raining.
As for campsites, just be aware that the Park Service is sequestering hikers into certain areas to mitigate impact on the environment; I can't say I really like any of them. If I had to give you some possible "avoids" I'd list 1) Mystic Lake, as it is well below the lake in the forest with no views 2) Mowich Lake, this is a drive in campground with the attendant noise...stay at Eagle's Roost (1.5 miles away) if you have to overnight in this area 3) Sunrise Camp, too close to Sunrise so lots of noise from "drive-in" campers. The rest are OK. As I mentioned above, Summerland is very nice but an easy day hike so it tends to be full in the summer. The section between Longmire (cw) and Mowich will give you the most seclusion from day hikers.
#4- If you have done the trail before, I'd love to hear your itinerary. We plan on doing 9 or 10 days, and would like to have an example or two of what worked for a leisurely pace. I usually only spend 1 or 2 nights when I do this trail, but here's a link to a nwhikers thread where people were posting their itineraries for last year and should give you a good idea for longer itineraries, including side trips and rest days: