Rapunzli Welcome to the forum!
You should open a new topic so that you're more likely to receive relevant replies. It also makes it easier for others with the same questions to search for your topic.
But to quickly answer your questions:
Rapunzli I want to have detailed control, whether an app may use WiFi or mobile net or nothing. There are many apps, which do not need any network connection.
You don't need firewall apps as GrapheneOS implements the Network permission toggle which can be found from Settings > Apps > [App name] > Permissions > Network or by holding on the app's icon and pressing App info.
Rapunzli So my question is, whether there is no need for an app like AdAway on GrapheneOS, because it would suppress unwanted ads on system level. Or does someone know of an app, which would be compatible with GrapheneOS?
The recommended approach is to choose a DNS server that blocks ad domains. Ad-blocking apps are unrecommended.
Rapunzli Third, I would like to control the charge levels of the battery in order to increase battery life. The only apps, I know, demand root. So my question is, whether there is an app, which would control battery charging and is compatible with GrapheneOS.
I don't have the answer to this so I can't say anything. You should open a new topic 🙂.
Rapunzli I am working as a practitioner for mental health and there are some very sensible data from my clients on my phone. If the device gets stolen or misused, I want the device being wiped for security.
GrapheneOS has a duress PIN/password feature which securely wipes the phone, unlike incomplete third-party implementations which allow interrupting the wipe. This might not be what you're looking for, but Google Pixels have some of the strongest hardware implementations against bruteforcing. You can also enable Auto reboot.