How do I prevent my PPP connection from disconnecting when it's idle?
I found a solution. The file preferences.plist inside /Library/Preferences/SystemConfiguration contains the configurations for the various network interfaces.

Looking at the file with a plist editor I found the following data under Root / NetworkServices / ... / PPP:



After I've modified the Disconnect... related values from 1 to 0 with BBEdit, the automatic disconnect has been successfully disabled.

However, after another ~40 minutes, the connection was dropped with a different message saying that the connection was dropped because no (phone) number was given. That made no sense to me, because when I manually reconnected afterwards, it worked again. So, there's still something wrong with my setup but the idle disconnection is not it any more.