For vanilla/raw/stock (or however you want to refer untouched) Atmosphere, they go in the kip_patches folder.
Kosmos is a package that includes Atmosphere but uses a different bootloader than the one that comes with Atmosphere; Atmosphere is designed to be used with Fusee but Kosmos bundles it with Hekate. This doesn't affect the functionality of the CFW but Hekate is better than Fusee since it has more tools and can load other CFW unlike Fusee. Not to mention that Kosmos comes with all of the stuff you would normally have to add manually to Atmosphere such as patches, the homebrew menu, and some useful homebrew applications.
The exploit is temporary as most exploits on newer consoles are. Whenever the console is restarted or turned off, the exploit is lost and needs to be re-initiated. Sleep mode doesn't count as the console being off.
There are 2 categories of keys: console-specific and public. The only console-specific keys are the biskeys which are only used for decrypting the NAND via hackdiskmount. The public keys are, as the name implies, shared across all consoles but cannot be shared for legal reasons. Instead various tools are used to dump them from an individual's console so people can obtain them legally and not have to look them up on the Internet. The public keys serve a wide variety of functions from converting between .XCI and .NSP format, unpacking .XCI and .NSP files, and dumping games in .NSP format.