No, adjectives are never capitalized unless they are the first word of a sentence.
yes
Yes. America is always capitalized -- even if it is being used as an adjective.
Yes, "Cherokee" should be capitalized when used as an adjective to refer to the Cherokee people or their culture.
Yes. Any time "Greek" is used as an adjective or a noun, it should be capitalized.
Yes, because it is a proper noun and the name of a specific church, "Presbyterian" is always capitalized. It can also be used as an adjective, as in "They follow the Presbyterian methods and rules."
Always
Yes.
Neurological should be capitalized when it is part of a proper noun or at the beginning of a sentence. For example, in "Neurological Disorders." However, when used as a general adjective, it does not need to be capitalized.
Not necessarily.
Yes, "Western Europe" is capitalized because it refers to a specific geographical region. When used as a proper noun to denote that region, both "Western" and "Europe" should be capitalized. However, when used in a general sense or as an adjective, such as "western European countries," it may not be capitalized.
The capitalized word used as an adjective is "North" in "North American." In this context, "North" describes the geographical region related to America, indicating it refers to the northern part of the Americas. The other words do not function as adjectives in this context.
If referred generically as you did in your question, no. But, a specific one, such as Facebook, should be