I attended the Salzburger Festspiele in 2014. It was very special, and I was "introduced" to legendary conductor Nikolaus Harnoncourt, who became my favorite conductor! You will enjoy it. As an aside, I stayed at the Gästehaus im Priesterseminar. No a/c. Hot! But paid something like 67 euro per night for a single room, as did a friend of mine (though that was a decade ago).
I think Königssee at Berchtesgaden is grand. It's a good choice if you would like a fairly easy hike with fairly big views. You can take a boat ride across the lake, hop off at the Salet stop, walk to Obersee, hike around the right side of Obersee to its far end, and then ascend to a gorgeous alpine meadow surrounded by peaks. At the end of the alpine meadow is Röthbach Waterfall, and the meadow is shared with cows doing their summer grazing.
I have spent some time in the lake districts but not the "popular" lakes. Attersee. Mondsee. Both are nice to visit if you have reason to be where they are, but they are not Königssee. Others can chime in on the more popular lakes in the Salzkammergut.
Sounds like a great trip! Hope you enjoy your time in Salzburg!