Do app developers know when users disable notifications? Some iOS apps annoy me with push notifications that I cannot customize inside the app, so I go into iOS settings and turn off notifications entirely.

But I wonder, do app developers get any metrics on this?  I'd like them to be aware that their customers are turning off notifications, so maybe they could improve the app.
They can't tell when a user has turned off notifications.  Apple doesn't share that info with developers.

The developer can tell if a notification was dismissed or a button was pushed.

Those are types of notifications that you, the user, must take action on.  There's another type known as silent notifications that don't require you to take action.  These are used by Watch complications, VOIP apps, etc, and they do things like put badges on the icon.  These will be delivered to your device whether you have notifications enabled or not.  

A developer is 'promised' by Apple that these notifications will be delivered so the developer knows if action was taken because of that (i.e. you read an email because an icon badge told you there was a new message).  

The best thing to do is notify the developer directly that you don't like what they're doing.  Honestly, some developers don't care and won't change.  In that case, stop using their app and find something better suited to your needs.