Sql-server – Notification mail not sent on failure

(1) Yes, I usually have the backup process by itself. I would not be doing much of anything during my backup time if I can. You might have it take the backup and then do the update on stats. As it sounds it seems you are running two jobs (1 for backup, 1 for update stats) at the same time?

(2) Do you copy the backup to tape or other disk storage? If so then I usually clean up the files before I create new backups locally. If not, then if I am scraping for storage space I would consider zipping the backup file prior to the new one being created. (That is if you cannot enable compression on the backups, as @Simon suggested, this would save some space as well.)

Have you created a contact group in Outlook or a Distribution group in Exchange? (I'm assuming you are using Exchange?)

The Outlook group is just a container that lets you send emails to many people from your personal machine. The SQL Server is trying to send to a group that exists only on your Outlook and not on the Exchange server which is why single email recipients work but not the group.

I would suggest setting up, or asking your email admin to setup a new SQL support distribution group on your mail server.

or

You can send to multiple single email addresses by separating the address of the recipient with a ;