Am I missing something? There doesn't seem to be a way to set an alert to notify anyone that there are X-number of interactions waiting in the queue. Whether inbound or callbacks (particularly callbacks). Also, are there any options to have a persistent on-screen notification above the notification tray area (not in-browser)?

There is no functionality in alerts to do that right now. It is one of the ones that has been looked at.

No option for persistent on-screen notification, no, except for Time in Status alerts.

Contact Support to make sure your need is registered, so Product Management is aware of demand for those features.