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

It’s for the Play Store, which accepts 512×512 high-resolution icons:

• High Resolution Application Icon (Required):
  • Use: In various locations in Google Play.
  • Specs: 512×512, 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.