Any correlation between tempdb and sysmail? Is there any correlation between [tempdb] getting full and subsequently  db_mail disabled ? [This needs sysmail_start_sp to be executed in msdb]
Yes there is correlation between tempdb and db_mail.
Ref:
https://technet.microsoft.com/en-us/library/ms345368(v=sql.105).aspx
Database mail user service broker which in turn use tempdb.

Service Broker helps developers build asynchronous, loosely coupled
applications in which independent components work together to
accomplish a task. These application components exchange messages that
contain the information that is required to complete the task. Service
Broker explicitly uses tempdb for preserving existing dialog context
that cannot stay in memory. The size is approximately 1 KB per dialog.
Also, Service Broker implicitly uses tempdb by the caching of objects
in the context of query execution, such as work tables that are used
for timer events and background delivered conversations.
Database
Mail, Event Notifications, and Query Notifications implicitly use
Service Broker.