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, "Presbyterian" is capitalized when referring to the Presbyterian Church or its members, as it is a proper noun. However, it is not capitalized when used as a general adjective to describe a style of church governance.
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.
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.