First, with that kind of time, Baden-Baden, a place Mark Twain claimed you simply MUST go to before dying, is a must do. I would try to space it as close to the middle of he trip as you can. It is a recharge moment. 2 full days and nights is good for Baden-Baden. Fredricksbad (sp?, I will check). Go for the spa, which is about $30 or so for about 5 hours of oh my god. My wife and I loved it when we wen several years ago. Go on a nude day. Nothing weird, it is just very freeing. There is a kids water-spa-kinda-park next door.
Good spa, food, etc. love it and will go back next time I have time...
All right, I have had time to do some basic thinking about this.
Driving days don't suck. In fact, they can be really good days. They are a chance to leisurely stroll (see some of my coming change suggestions) through areas few foreigners (or at least fewer) ever go. With a car you can wander into a a tiny home shop on the side of some twisty turny road to buy cheese or honey and end up touring their family-run operation. One of a few versus one of 50 or 100s a day. I have had very intimate experiences (I work with bees and huts) that I think driving in car made possible. As far as parking I will say this: I have carried (ie lugged) luggage from a station to a hotel and I have found hotels/houses that provided parking or I have found parking on the street. I prefer the latter.
Now my suggestions...
6/23,24 - Hamburg
6/25 - drive 6 hours stay in Baden-Baden(same drive time but no border crossing or need to buy a sticker for Czech)
6/26, 27 - Baden-Baden
6/28 - drive 4 hours stay in Hallstatt (drive goes from 5 to 4)
6/29, 30, 1 - Hallstatt 3 nights (visit Salzburg)
7/2 - drive 5 hours stay in Venice
7/3, 4 - Venice 2 nights
7/5 - drive 5 hours stay in Cinque Terre (via Florence)
7/6, 7 - Cinque Terre 2 nights
7/7 - drive 5 hours stay in Saas Fee (drive goes from 5 1/2 to 5)
7/8,9 - Saas Fee 2 nights (car-free and around the corner, literally, from Zermatt; park right outside the city "gates" in a cute little tower)
7/10 - drive 3 hours stay in GMG
7/11, 12, 13, 14 - Gimmelwald / Murren/ Grindelwald 4 nights (day trip to Bern etc).
7/15 - drive 1 1/2 hours stay in Luzern
7/16, 17, 18, 19, 20 - Luzern 3 nights (Vierwaldstatersee, FraekMuentegg, Army installations on your way in from GMG)
7/21 - drive 2 hours stay in Appenzell/Ebenalp
7/22, 23, 24 - Appenzell/Ebenalp 3 nights (stay in one of the public huts, maybe a different one each night, carrying nothing but your phone, charger, wallet, change of under wear, water, and a day bag--my 3-6 year-old can/has done it) MAKE RESERVATIONS, even if you do them on the ground, but this is a busy season!
7/25 - drive 1 1/2 hours stay in Zurich
7/26 - Zurich 1 night, (fly home from here, stay at Apart-Hotel, www.aparthotelzurich.com/ 8 minute free shuttle ride to the airport after shower and tasty early morning breakfast)
Good trip wherever you go! And June/July is BUSY SEASON, RESERVATIONS can put your mind at rest and get the ONLY AVAILABLE parking place when it comes to it :)