American is a proper noun. It can also be an adjective, as in "American cheese." A rule of thumb: proper nouns are capitalized and common nouns are not capitalized.
Fontina should be capitalized but not cheese.
Yes because it is a specific type of cheese.
They are not proper nouns. They would not be capitalized in a sentence.
Yes, only the proper name "Swiss" is capitalized; cheese is a common noun.
Yes. The word American is always capitalized in English, as are other nationality adjectives. Although a case could be made for "American cheese" if "french fries" is lowercased.
Yes it should be capitalized.
Yes, "Swiss cheese" is typically capitalized because Swiss refers to the type of cheese rather than the country.
Yes, the Mozzerella should be, the "cheese" needn't be.
Yes, "Roquefort" is traditionally capitalized as it is derived from the region in France where the cheese is made.
No, "romano cheese" should not be capitalized unless it begins a sentence. It is a type of cheese named after the regions in Italy where it originated, but as a common noun, it does not require capitalization. However, if referring to a specific brand or product name that includes "Romano," then it would be capitalized.
Yes, "Smoked Gouda" is usually capitalized as it is the name of a specific type of cheese.