Every single thing you just mentioned both occurs and doesn't occur with both web and native apps depending on what the app was built to do.

- Web apps can require internet

- Web apps can not require internet

- Native apps can require internet

- Native apps can not require internet

- Web apps can inform a server about what you're doing

- Web apps can not inform a server about what you're doing

- Native apps can inform a server about what you're doing

- Native apps can not inform a server about what you're doing

- You can unplug from the net, install desktop software (native app) and run it without it being able to send anything anywhere

- You can not unplug from the net, install desktop software (native app) and run it without it being able to send anything anywhere

- You can unplug from the net, install desktop software (native app) and run it without it being able to send anything anywhere until you plug back into the net

- You can unplug from the net, install desktop software (web app) and run it without it being able to send anything anywhere

- You can not unplug from the net, install desktop software (web app) and run it without it being able to send anything anywhere

- You can unplug from the net, install desktop software (web app) and run it without it being able to send anything anywhere until you plug back into the net

Source: I make offline web apps.