What is this Google Settings app that installed itself on my phone?
The Google settings app is being put their either by the Maps app, Search app, or the Google+ app.  

It's a unified settings/sign-in thing they just added in literally yesterday and allows you to manage the settings for all of the above (and probably more in future) from one menu.

http://www.engadget.com/2013/02/27/google-settings-app-sneaks-onto-android-to-bolster-g-sign-in/