Re dogs: in Mt. Rainier NP and North Cascades NP, it's OK to bring your dog on the PCT (no other trails, though) as long as he's kept on leash.
Depending on the time of snow melt, late July may still be quite buggy, especially on the plateaus that have lots of lakes: Sky Lakes, southern Three Sisters Wilderness, Jefferson Park, Olallie Lakes, Indian (Mosquito) Heaven, the section just north of White Pass (Mosquito Valley). Be prepared (head net, repellent, screened shelter for both you and your dog)!
The Three Sisters Wilderness and southern part of the Mt. Washington Wilderness can be really hard on your dog's paws due to pumice and lava abounding everywhere.