You are going to love this tour! The hubby and I did it two years ago, leaving late June and coming home early July. The weather was somewhat iffy. It rained a few days, and was sunny a few. Definitely pack a small umbrella and a rain jacket. A fleece layer works well underneath, especially since you're going to the Schilthorn.
Don't over think the shoes too much. One pair of tennis shoes or walking Merrills will be fine. Murren hiking is on trails, and unless it poured for days before, the trails will be fairly nice. We also took a pair of Teva sandals for the nicer weather days. I think you will find a lot of walking, stopping, then waiting. If you don't want to walk up to Neuschwanstein or Berg Eltz Castles there are buses for a small fee.
The hubby took zip off pants, although I don't recall him ever wearing them as shorts. He did wear jeans and cargo shorts, although Dockers will work well enough. Take a pair of shorts, just in case. Also, take swim trunks because the Caracalla therme water park and sauna in Baden Baden is a MUST DO!!! This is not the Roman baths where they put mud on you and scrape it off (it was closed), but a swimming/spa environment with jacuzzis, saunas, and swimming pools. Some people may be scared off because of the clothing optional part, but that's upstairs and out of the way, completely isolated, and totally separate.
We had backpacks as well, and were very happy we had them. It was much easier to walk up the stairs with our packs than to wait for the elevator with everyone else. Besides the walk from the train station in Trier, the walk from the gondola in Murren to the hotel is through town, and packs will be easier. Plus you don't have to worry about rolling the wheels into any one or anything. Go to REI and have them fit you. We are so happy we did, plus we practiced carrying them and filling them. In fact, we'll be taking them on the Athens and the Heart of Greece tour in June.
Handy tips: Plan for a concert while in Salzburg. You can scout this ahead of time, but there are so many the guide will be able to help you. We went to the Mirabel palace, and it was lovely. Also do the bike ride along the river, out the SOM gazebo. Even if you're not a fan of the movie, the bike ride is wonderful and the scenery is stunning!
If you have a chance, go to the bone-church in Halstatt, although that could interfere with the salt mines. We skipped the mines in favor of a lingering lunch over the beautiful lake and a meandering walk through the town. I think that was the only down time we took.
Schonbrunn Palace in Vienna is beautiful, and very easy to find using the subway. We bought our tickets when we got there, then took the time to walk up to the Gloriette for the most stunning views.