Rather is an adverb. It is used before an adjective (John's socks were rather smelly), another adverb (He was walking rather slowly) or a verb (I like pop music but I rather like listening to...
No, it takes an adverb to do that. In the phrase a red car, the word red is clearly an adjective. In a big red car, we have two adjectives, both describing the noun, the car. So a bright red car also...
Yes, a proper adjective would be the adjective that is derived from a proper noun and used to describe another word. Examples are "I like Asian food." where Asian is the proper adjective describing...