The short answer is no, especially in the summer, most everyone speaks English. Nick knows some German and will help translate. However, we highly recommend brushing up on beginner German. Duolingo or Babbel are great for this.
We will be taking the train to Innsbruck from Munich. Nick will have rented a car and we will need one more car for a couple of the activities (which can be rented in Innsbruck for a day or two). Parking in Innsbruck is extremely difficult but there are garages where you can store a rental car.
Both Germany and Austria use the Euro. Most stores accept credit cards, but we recommend getting some cash from a bank. Germany has more banks but we recommend researching which ones work with your bank in America without fees.
We will be staying in Innsbruck through July 31st and leave for Munich that morning. We suggest that you fly out of Munich (if you are planning on leaving from Innsbruck). This is the biggest airport near Innsbruck. If you are flying out in the morning we strongly suggest that you plan in advance your ride back to Munich. There are trains between the two cities but they take roughly 2 hours and have limited schedules in the early morning.