My app needs to be able to update itself. In order to do so, when I click on the app's update to start the installation, it should stop and restart after installation is complete. Can Windows do that for me somehow?
You can split your application into 2 parts:
main application and
updater. You can call the updater from the main app (and stop the main app), do all the update stuff, then relaunch the main app from the updater.