Proper
The word "cherokee" is not an adverb because it does not modify anything.
Yes, "Cherokee" should be capitalized when used as an adjective to refer to the Cherokee people or their culture.
"Little" is a descriptive adjective used to convey a small size or amount.
"Long" is an adjective, not a noun. It describes the length of something.
The word "haggard" can function as either an adjective or a noun. As an adjective, it describes someone looking tired and worn out. As a noun, it refers to a type of hawk.
Raucous is an adjective describing the noun dancing.
Yes, "Cherokee" should be capitalized when used as an adjective to refer to the Cherokee people or their culture.
Proper
The word Cherokee as an adjective is a proper adjective; an adjective derived from a proper noun.
Each is an adjective that is also a pronoun. You can describe it as a 'distributing adjective'. See the link below for a description of this type of adjective.
New is an adjective.
The word "of" is not an adjective. It is a preposition.
The word "of" is not an adjective. It is a preposition.
the Cherokee traded with England
indefinite adjective. A+LS
The word "the" is an article, which is a type of adjective.
demonstrative adjective
The 2001 Jeep Cherokee has overhead valves (OHV).