Yes.
Yes, "Cherokee" should be capitalized when used as an adjective to refer to the Cherokee people or their culture.
Yes. America is always capitalized -- even if it is being used as an adjective.
In Spanish, months are not capitalized unless they are used at the beginning of a sentence or part of a proper noun.
You capitalize it when referring to a title. For example, Medieval Europe. It shouldn't be capitalized when it's being used as an adjective.
Yes, the S of Spanish should be capitalized, and not just when you are referring to the language. If for instance, you write the words "Spanish culture," or "Spanish rice," or "Spanish people," the S should also be capitalized in those cases. (but in Spanish, the first letter of a language is not capitalized; i.e. ingles, not Ingles.)
Yes, it is correct to capitalize "Should" if it is used in a title as it is a noun in this context.
You should capitalize a language when it is the name of the language or when referring to it as a proper noun, but not when used as a common noun. For example, "I am learning Spanish" (correct) vs. "I am learning how to speak spanish" (incorrect).
Yes, the term "Hispanic" should be capitalized when used in a sentence. It refers to people and cultures of Spanish-speaking countries or origins.
'US' must be captialized, but for the 'the' you should only capitalize it if it is at the start of a sentence (Eg: The US is a big country).
Yes, "Latino" should be capitalized when it is used as a proper noun to refer to individuals of Latin American descent. It is also common to capitalize "Latina" for women and "Latinx" as a gender-neutral term. However, when used as an adjective in a broader context, it may be lowercase, depending on the style guide being followed.
Yes, when it is used as an abbreviation. It should be---KP.
In Spanish, the seasons are not capitalized unless they are the first word in a sentence. So, both "otoño" and "primavera" should not be capitalized when used in the middle of a sentence.