Is it possible to install a different settings app than the one my phone came with? Specifically, I'd like the official Google one.
A Galaxy S3 is to highly customised by Samsung that you can't install a stock settings app. It will corrupt/brick your phone if you try so.
However you maybe could install the settings app of a galaxy s4 mini because it is almost the same. But I am not sure. do it at your own risk
Hope my answer helped.
There's an alternative method here if what you're looking for is greater control and access to settings not available through the default settings UI: Settings Database Editor
Word of warning: you can possibly brick your phone if you don't know what you're doing, so proceed with caution, back up your data, and thoroughly research any customization.
Through this app, you can directly access and modify the db values for your various device settings. This doesn't necessarily require root (might depend on device model & OS version), as you can grant some of the permissions through an ADB command.
EDIT: fixed link.