ODBC connection fail when running AD account for SQL Server services
As you have found out the SQL Server Service runs the backup but not the user giving the backup command. 

Changing the service user should be done using the SQL Server configuration manager which in the background sets correct permissions for the service account, all of which are documented on MSDN, giving the user running the service Administrative rights on the computer will make it possible to run the  service but is not a good security practice. 

You have to be sure to run the correct version of the Configuration manager so try running MMC.exe and adding the correct snapin for your version of SQLServer so check that first and if it failes reregister the snapin by running in a command prompt the following command

mofcomp "%programfiles(x86)%\Microsoft SQL Server\100\Shared\sqlmgmproviderxpsp2up.mof"

If this failes, use the information in the msdn article to manuallyl set the correct permissions.

Your SSPI context error is due to incorrect SPN for the sql server, The SQL server creates a SPN automatically when  you start it and the domain admin has permissions to to that which fixes the SSPI error. After you change the service account for the server to a normal user download the Microsoft Kerberos configuration manager for SQL Server and make sure to create the correct SPN