Is american/American a proper noun, and why?
Yes, "American" is a proper noun, a word for a specific nationality. A proper noun is always capitalized.
When American refers to a person, it is a proper noun just like John, Mark, Sarah or Emily. If you can substitute American in a sentence with John, for example, and have the sentence still make sense, then American is a proper noun. In the sentence, "An American walked into the room", you can substitute American with John and the sentence will make sense and still keep the same general meaning. Here are some examples of it used as a proper noun.
Americans like pizza.
The Americans got off the plane.
There are a lot of Americans in Paris.
American can also be used as an adjective and would then, of course not be a proper noun. Here are some examples.
New York is an American city.
I have an American passport.
American money is green.
The word "American" is also a proper adjective, a word used to describe a noun as of or from the United States of America the continents of North or South America. A proper adjective is always capitalized.