Disabling the iPhoto Photo Stream App badge icon without disabling shared streams

To solve the problem you state, Dropbox would allow you to upload all the photos from two devices to one account (or two accounts) and then set up folder actions on a Mac to combine, import and delete the photos from the dropbox storage.

There isn't any "mix these two" photostreams function in iPhoto or Aperture and the web interface doesn't have any functionality to match that as well.

There is a way to push photos (manually) to a shared photo stream, and then both of you would see the same content in that stream, but it's designed for one person to push - not for several to collaborate and publish together.

The pragmatic solution is to set a calendar reminder once a month for both of you to export the pictures. I let iPhoto automatically import all pictures from my Photo Stream and it conveniently makes a November 2012 Photo Stream event and I export those original photos to my archive drive that sits on a shelf other than when it's time to archive. Rather than archiving (or perhaps in addition to), you would then do the steps needed to export and import the batch of photos into the library you chose to contain both pictures.

This both isn't automatic and also means that photos will be potentially duplicated if the library holding the merged content is different than the library that received the stream from iCloud.

No - the normal path is to use the Notifications prefpane to turn off notifications, but this would not help, since Spotify is not listed there.

I'm also very annoyed by Spotify's constant flood of notifications. I wonder if there was a way to drop all the social-related network traffic that goes into Spotify and somehow nip these notifications at the bud.