Is there any way to stop a SQL Azure DB so that it doesn't charge anything towards our account? I don't want to delete it, just while in testing and it's not being used than we set it to "stopped" like we can do with websites, cloud services, and VMs.
They won't allow it. So the billing will continue for your Azure Database starting the day you create it. There really is no way to pause / stop billing for your Azure SQL Database.
Microsoft's official answer appears to be "Yes, you can export your database. Delete the Azure SQL database and that will pause billing. Then when you need it you can create a new database and import your previously expored DB."
I don't believe this is acceptable as an answer for "Allow me to temporarily turn off SQL Server to save on my billing"