SQL Server – NT AUTHORITY\SYSTEM Cannot Access Database During Backup

Since your network drive is available most of the times and the backup job only fails randomly, I would first check the network. It could very well be the network issue. I would also look at the intervals the backup job fails, to see if there is a pattern to it.

Coming to restarting the logshipping part - the easier method is to restore the full backup and the standby server and reconfigure. But it depends on the size of your database, resources available etc

It looks like MS has seen this already. Have you tried running SSMS through Right Click - Run As Admin?

http://connect.microsoft.com/SQLServer/feedback/details/198005/not-able-to-access-the-database-model-under-the-current-security-context-microsoft-sql-server-error-916

Also are you logging in as a sysadmin to SQL Server? Do you have create DB rights?

Edit: Just saw your original post updated. It looks like you are restoring to the wrong version for sure. Consider yourself lucky if that is the issue!!

Do a SELECT @@SERVERNAME. What does it return? Is that the right instance? Probably not (hopefully). Try your restore script only after SELECT @@SERVERNAME returns the right info. Also you might want to use OSQL where you have to state the server name and creds before connecting, that way you know you're at the right one instance.