What is the behavior of SQL Server Maintenance Plan - Clean Up task for deleting old backups
Does the expiration policy on the backup task "lock" those files from being deleted before they expire?


No. The Backup set will expire setting indicates when the backup set can be overwritten by another backup. It does not cause the file to be deleted, nor does it prevent the file from being deleted. 


  In essence, could I set the cleanup task to delete files older than 1 day, and yet not delete backup files that haven't yet expired?


No. The cleanup task will delete the old files. And any backup sets within those files will be gone, regardless of their expiration setting.


  If a FULL backup is deleted, then all the differentials are useless, so they should be deleted as well.


You are correct that the differential backups are then useless. But the cleanup task that deletes the full backup doesn't know that, so it has no way to delete the differentials. You just have to wait until the cleanup task runs the next time, and the next time, and the next time, and it will eventually delete the differential backups, one at a time as they reach the cutoff age (7 days, in your example).