If you wanted to use "far west" as an adjective, you can just say "far western." This is analogous to the adjective far eastern or Far Eastern.
But the the setting of Red Dead Redemption and dozens of westerns (as a noun) such as 3:10 to Yuma and The Good, the Bad, and the Ugly and books such as Death comes for the archbishop is not
"the far west"
But the "American West" or, if American is understood, simply
"the West" or "the old West" or "the wild west".
If one wanted to include Canada and/or Mexico, one could say
"The North American West."
The adjectival form of any of the above would be simply to change West to Western.
But western as an adjective with regard to these geographical areas, is largely used for books and movies, but not computer games. For instance, if I walked into a computer game store and asked if they had any 'westerns' or 'western games', they would probably ask me what I meant.
So one would probably have to settle for
Red Dead Redemption is set in the (Old)(North)(American)(Wild) West. Since Mexico is important in the game, it would be better to reference the North American West, or simply, the Wild West, since that term can include Mexico.
Also, since "western" can refer to both the western hemisphere and to western civilization, one would need to make sure that it is clear from context what you mean by "west". I expect you meant this by using "far west." But in English we simply don't use this to refer to the American West.
Also note that the current "Western USA" or the current American West might exclude some states traditionally included in the Old West or the Wild West. This is because these states are more central geographically. Although the city of Fort Worth, Texas refers to itself as "Where the West begins" and St Louis, Missouri is known as "the Gateway to the West".
Cook book is not two words but one: cookbook.
Do you mean a library that contains only cookbooks? You could say it is a library that specializes in cookbooks. Or a person could say that they have a library of cookbooks or a cookbook collection. I would tend not use "Cookbook library."
In many cases you can just use two or even three-word nouns as an adjective:
mobile phone app
dining room table centerpiece
ice cream scoop
beer can collection
online league draft
horror movie fan
computer game store
Mexican food restaurant
You can, in fact, make longer strings of multiple-word nouns used as adjectives:
Ice cream scoop
Ice cream scoop rinser
Ice cream scoop rinser parts
Ice cream scoop rinser parts catalog
Ice cream scoop rinser parts catalog page
Ice cream scoop rinser parts catalog page number
Ice cream scoop rinser parts catalog page number font
Ice cream scoop rinser parts catalog page number font size
Ice cream scoop rinser parts catalog page number font size finder
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Ice cream scoop rinser parts catalog page number font size finder fee war...