Do I need a Eurorail ticket and a Swiss pass?
Difficult to say without knowing you itinerary in Switzerland, but I would say from experience that, for the parts of your journey in, or partially in, Germany, you would save by using Savings Fare tickets, regional offers, and point-point tickets. In my experience, it is difficult to make a rail pass pay.
For instance, with advance purchase, you should be able to get a Savings Fare ticket from somewhere in Switzerland to Heidelberg for less than 100€ for two people. I would spend at least the first night in Switzerland somewhere near Zurich and travel to Heidelberg at least one day later. Two reasons, one, you don't want to take a long train trip from your arrival airport the first day when you are jet-lagged from the long flight and time zone change. Second, although advance purchase tickets can save you a lot of money, they are train specific and can't be exchanged for later tickets on the same day as you use them, so if your flight is late, you can lose the ticket. Better to go to one of the places in Switzerland you are planning to see anyway, somewhere close to Zürich, and then use the Savings ticket on a later day when you know you can make the specified train.
Heidelberg isn't that far from Stuttgart. It might take an hour longer by regional train, but you can use, for instance, a Baden-Württemberg-Ticket at 30€ for two people. That ticket allows you unlimited use of regional trains in 2nd class in Ba-Wü for the entire day, so you still would have the flexibility of a rail pass. And the same (31€ for two people, unlimited travel on regional trains) would apply to the Bayern-Ticket for a day trip from Munich to Füssen. And the Länder-Tickets (as they call the Ba-Wü- and Bayern-Ticket) are valid for all local transportation by bus, tram, and U-Bahn, not just S-Bahn, so it would cover you to the Hbf in Munich and from Füssen to Hohenschwangau by bus, which the rail pass would not cover (rail passes cover S-Bahn but no other local transit). All of the so-called regional offers (Länder-, QdL-, and Schönes-Wochenende-Tickets) are one price whether purchased in advance or just before you use them (they do cost 2€ to purchase at a counter rather than from an automat).
For Stuttgart to Munich, two of you can do the trip on a direct ICE in just over 2 hours with an advance purchase, train-specific Savings Fare ticket starting at around 50€/2. With a Quer-durchs-Land-Ticket (52€, weekdays) or a Schönes-Wochenende-Ticket (50€, weekends) on regional trains, it will take just over 3 hours and you will have one change in Ulm, but you will have the complete freedom of unlimited regional trains in both Baden-Württemberg and Bavaria, so you could stop anywhere along the way if you wanted to.
Lastly, back to Switzerland, with a Savings Fare ticket from the Bahn, you could go from Munich to Zürich by Eurocity train for about 50€ or by non-stop Bahn IC bus for 40€. Or, you could go by bus or train to Zürich and then on to Interlaken Ost by train for less than 100€.
So, depending on you Swiss routing, it might be best to use local tickets in Switzerland and what I've suggested in Germany or a Swiss pass and what I've suggested for Germany. I know what I've told you is complicated and there are a few more details, like what kind of Saver Fare tickets (regular or Super) to get and how to order your tickets. You need to get familiar with using the Bahn schedule and fare webpage.
There are others on here who can advise you better about your Swiss travel. When you have a better idea of your Swiss itinerary come back and there are people here who can help you decide what way to go.,