You need to cut Austria.
I am not familiar with the Black Forrest. You can look up train schedules from www.bahn.de, then click on Deutschland along the top to get the English Version. With your limited time I would head to Wengen or Murren in Switzerland for the Alps. It is about a 6 hour train ride from Stuttgart.
Murren has several great walks for kids, one is the Kid's Adventure Trail, primarily down hill to Murren. You walk through fields, pastures we saw goats and horses. It has activities along it that may interest kids like a small zip line. It passes over a very cold mountain stream. You can take the funicular up from Murren to Alpenhubl, where there is a great kids play ground with tunnels and zip lines and slides. Our 7 year old loved it! After an hour, she cried when we left! Another trail is the North Face Trail, longer and more hills, but it has the BEST scenery! That is why you go to the alps.
You can break up the walks with a trip to Schitlbahn or a train to the Jungfrau
If you rented a car and drove through the Black Forrest and turned you car in at Bregenz it is roughly a 6 hour train ride but with many more connections.
May 26 Stuttgart, Arrival/Recovery Day
May 27-May 30 Black Forrest
May 31 Train to Murren.
June 1-4 Murren.
June 5 Train to Paris, a seven hour train ride with four or five train changes.
June 6 Eiffel Tower, Reservations Required or you stand in a very long line.
June 7 Paris, something else that suits your fancy. Walk to Notre Dame, St Chappell, Sacre Coeur, Arc de Triomphe, Champs Elysees. Enjoy croissants in the morning an grab a gelato to see the Notre Dame after you have climbed to the top and seen the huge bell while admiring the flying buttresses.
June 8 Paris then noon train to Stuttgart, 3 hour train ride with no connections to 5 horus with four connections.
July 9 Fly Departure day.
I added Paris because you said you wanted it, I would skip Paris because it is taking up two days just to get there and back. I removed a day from Switzerland and added it to Paris so Paris would not be so rushed. Even with two days it is rushed. You only have time for a highlight tour and that is it.
I would skip Paris and add time to the Black Forrest and extend it to Fussen and see Neuschwanstein and some of the older cities then add another day in the Alps then a train to Stuttgart on July 8th.
Your choice, but I know having bucket list items. I hope this helps.