Ok...let's take this point by point:
1. Map: The best map I have found so far is National Geographic's map of the park. It shows everything in great detail, especially the trails you need to use to reach many of the streams. LRO has them.
2. Flies to use: There are a lot of flies that would work well in June, but the color to key on is yellow - yellow stonefly patterns, or a yellow Parachute Adams will work well.
3. Where to fish: There are a lot of choices...if you're staying in Gatlinburg, the West Prong of the Little Pigeon is hard to beat, and very easy to access; just pull into any of the "Quiet Walkways" along the Newfound Gap road (U.S. 441) and hike down to the stream. Just be careful wading this stream - it has huge boulders, and is particularly slick (at least to me)...I usually bash my shins at least once on this stream.