> I am not sure what the solution is to these kind of apps. Maybe I can find a special mail app, that only shows you important emails. If I had something like this I think I would just be over the moon with this setup.
I have always had email notifications turned off and I was always missing important emails, especially from people I cared about. I finally figured out the solution. In Gmail (only tested on Android, can't speak for iPhone) I created a label called "notify". I then created filters for specific emails and words that apply the label. You can turn on notifications in Gmail (for Android at least) for specific labels. That's it! Maybe someone else can confirm that this can be done on the iPhone Gmail app? or something similar