but that defeats the purpose of only upgrading one package, right?

I mean, running Arch, a distro that sort of prides its self on being quite cutting edge also defeats the purposes of only upgrading one package...

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Running Pacman -S firefox does not work because the package lists have not been updated.

Running Pacman -Sy firefox works as expected because the package lists HAVE been updated. It is bad because if firefox(new version) requires dependency(new version) pacman will only upgrade firefox and not the dependency and your firefox will break... Sometimes it will work just fine for a reallllly long time. Sometimes it wont.

No, Pacman is not smart enough to realize that your dependencies are out of date. It just knows that the dependency is installed and doesn't care what version it is. If you ran pacman -Syu and upgraded everything, like everyone says you should none of this would matter.