Hi,
Thank you very much for the details!
1. Do you mean displaying the live results under the divi search? That is not possible unfortunately.
2. It might be only the browser emulator doing it. On a mobile device it works all right: https://i.imgur.com/GpdDIkS.png
3. Well, for that you need two separate search bars. It is possible to hide the plugin on mobile/desktop views via this option. You could make one visible on mobile only, the other only on desktop.
4. Changing the z-index here should do the trick: https://i.imgur.com/ZLovIfw.png
