With only seven days, and preferring to hike vs. tour cities, I think you have a pretty good break out between Salzburg and Mittenwald. But the area around Salzburg itself is not mountainous, it's more rolling hills. And most of the hiking is outside of the city. I love Salzburg, it's a really beautiful place to stay, but mainly for the city. Innsbruck would give you more of an outdoors Tyrolean experience. It's right down in the Alps, and about equal distance from Munich as Salzburg.
Likewise, I prefer Garmisch-Partenkirchen to Mittenwald, but that's because I used to ski, and they have the better runs (it was the site of the Olympics back in the '60's) . The fact that it was used as a major recreation area by the US military for decades makes it really well geared towards American visitors. And, if the weather is clear, a trip up the Zugspitz is better than any of the castles around Fussen. You can spend a whole day up on the top or ride up and walk down if you want to do something different.
One other site I highly recommend would be Chiemsee, a beautiful, large, lake between Munich and Salzburg. If you go to Salzburg it would be worth spending a day here. Lot's of easy walking trails around the lake with incredible scenery.
One thing you should not worry about is finding an "Old World" atmosphere. It's everywhere and that's the way the locals like it. Yes, there are many modern buildings and facilities, but part of the charm of Bayern and Tyrol is the way it looks and is maintained. Part of the reason they keep it that way is to encourage the hordes of tourists. If you drive, or even just do some hiking, you're going to find small towns and villages that are exceptionally scenic. Don't hesitate to just stop and spend some time in these smaller places, it's always worth it.