Thanks for sharing Andreas.
I see no reasons not to add opt-in badging for those that would like it when we have time to engineer it. As Ryan said, I too want Cake to be a place to make people happy, and opt-in badging helps with that.
We have the first two bullets you list. Unfortunately, we do not have an Android app yet, but we do treat iOS notifications similar to email notifications in Settings. You can toggle on and off categories of notifications.