SQL Server Agent initiating self-termination
Since you recently upgraded a few months ago from SQL Server 2005 EE to 2012 SE, you should save off and clear your event logs because SqlAgent may crash when it sees Event Log IDs from an older build of SQL Server. For example, the Agent may crash because it reads an event logged by a SQL Server build earlier than the current build, which only contains data with ONE insertion string and the agent will try to format it using the latest resource information that requires TWO strings.  This would lead to a brief memory leak just before it shuts down since a buffer is allocated for the string that it is trying to print into when the crash occurs. 

As far as the agent service not listed in configuration manager, there are multiple possibilities:
https://support.microsoft.com/en-us/kb/941823

Ensure you are running 2012 Configuration Manager

You can try rebuilding the WMI Repository:
https://blogs.technet.microsoft.com/askperf/2009/04/13/wmi-rebuilding-the-wmi-repository/