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.
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, 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."
If referred generically as you did in your question, no. But, a specific one, such as Facebook, should be
Yes, it should. As you can see from the 'Terms' on the Facebook site, 'Facebook' is always capitalized. The logo has nothing to do with if it should be capitalized or not. Facebook is a proper noun, therefore you capitalize it. It's odd that people would think otherwise, because you wouldn't put something in all caps just because the logo is in all caps, would you?
Jr is the abbreviated adjective junior, capitalized because it is used as part of someone's name.