Yes, "Japanese" should be capitalized when referring to the nationality or language (e.g., Japanese culture, Japanese cuisine).
It is a proper noun, and thus should be capitalized.
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.
No, proper adjectives do not always have to be capitalized. However, they are typically capitalized when they are derived from proper nouns.
Yes. Any time "Greek" is used as an adjective or a noun, it should be capitalized.
"Japanese" is capitalized incorrectly. It should be lowercase in this context, unless it is part of the proper noun for a specific group or organization.
yes
Yes, it is a proper noun or adjective.
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.
Lisbon is the name of a city, a proper noun which should be capitalized. There is no English adjective for a person or thing of or from Lisbon, but the proper adjective in Portuguese is Lisboeta.
No, "nineteenth-century" should not be capitalized unless it is at the beginning of a sentence or part of a title.