I'd be more than happy to answer your questions and I hope they are helpful as you decide what you may want to pursue in the future.
Yes, there is a lot of overlapping classes between Material Science and the other engineering majors. I took about 2 or so years of college mathematics, 2 years of chemistry, and 1 year of physics. I also took engineering courses in things like statics and dynamics, computer aided design, and technical writing. Specifically in the Material Science and Engineering major, there are courses for material analysis (microscopy, spectroscopy), crystallography, and classes on specific types of material like polymers or composites.
The particular area of Nike that I work in does the testing of materials, but there are other areas in the company that work on the material selection and design. Creating new products and designs is generally up to several other groups, but it is all very collaborative -- if I come to a meeting with test results for a material, that will affect the decisions made for products and designs.
Within my department there are a lot of different types of engineering backgrounds -- mechanical, chemical, bioengineering, electrical... my boss has a degree in aeronautical and astronautical engineering. When you extend the consideration to the people I work with in other groups, then there's even more backgrounds (both engineering, highly technical or not technical at all).
The amount of time that I spend working changes all the time depending on workload -- generally speaking we are always very busy so it's quite common to be working over the typical 40 hrs/week. However, there is also a lot of time within there to balance things out -- going out to lunch with coworkers, taking interesting classes, going to the gym on campus, etc. And we have a good work-life balance; I've never had an issue with taking time off to take care of my family or to go on a planned vacation.
At this point in your career path, any experience you get will be helpful. Finding an internship which pertains to engineering will help you decide if that's the right path for you, but even working at a seemingly unrelated place like a summer camp or a restaurant or department store would give you vital skills in customer service and communication. I would ask your teachers for ideas of where or who to ask in your local area that may be looking for interns.
Hope that helps!