What does the ic_launcher-web.png in my project root do?

This is the icon for the google play store. You need to provide a 512 x 512 icon.

There is a little bit of an explanation here on this page:

http://developer.android.com/guide/practices/ui_guidelines/icon_design_launcher.html#size

It's for the Play Store, which accepts 512x512 high-resolution icons:

• High Resolution Application Icon (Required):
  • Use: In various locations in Google Play.
  • Specs: 512x512, 32-bit PNG with alpha; Max size of 1024KB.

-- Google Support

(A more tutorial like explanation can be found here.)

It is not used in your actual app or the launcher, so it is not packaged in the APK.