Usually it's a noun, but it can be used as an adjective.
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No, it is not an adverb. Game can be a noun or adjective, and more rarely a verb. The adverb form of the adjective game (willing) is gamely.
difficult
Well you need to pay attention it is elastic
Playful is the adjective form of play.
No, the nouns 'skill' and 'game' are common nouns, general words for any 'skill' or any 'game' of any kind.A proper noun is the name or title of a specific person, place, or thing; for example, the National Skills Development Corporation (NSDC short name Skill India) or "Game of Thrones" TV series.The word 'game' is also a verb and an adjective.