In the Navigator app we find a lot of states as below.
I tried to modify for example <state name="supportingDataDownloadEnabled" type="boolean" value="true" id="v5egx"/> and was expecting that the checkbox in the app was set to True. However nothing happened.

Can anybody tell me what the states below do and how to set the checkboxes of the Navigator app?

The AppController.js is setting the supportingDataDownloadEnabled state variable to false in the applicationInitialized logic so the default value on the state in the app.xml won't matter. You could override this in the AppCustomization.js using another event. 

I understand the desire to have this defaulted, but I would be very hesitant to modify the navigator application. This application is fundamental to how mobile operates and can't be cloned. Every patch will overwrite your changes and need to be reapplied properly. A mistake during the build process can negatively impact how all of Maximo Mobile operates. Some examples of potential issues:

1) using the wrong version of the config tool. We bundle the graphite framework in the config image and going higher or lower can cause it to function differently than intended on that version. I've seen customers use a 8.11 image even though they're on 8.10 and I've seen them stick with the 8.10 base image even though their Maximo is running 8.10 IFIX 06. 
2) publishing the wrong app.xml/controllers. The config tool works from your local workspace when you have files. I've seen customers upgrade their Maximo environment and republish their app without realizing they published the old version of the app.xml/controllers because they didn't re-download it from the server. 

These types of things can lead to issues that are extremely had to identify because you're in a state that normally won't happen.