Backup Database (SQL 2000 Compatibility) on SQL Server 2005 with CHECKSUM, but Page Verification TORN_PAGES Detection I am trying to backup a 250GB database on SQL Server 2005 running in SQL 2000 compatibility mode.

The page verification is set to TORN_PAGES detection.
So, far the backup has been corrupted when we try to copy the backup to an external and restore on a different server.

The error I got in the past was as follows:
System.Data.SqlClient.SqlError: RESTORE detected an error on page (0:0) in database "MMS.mdf" as read from the backup set. (Microsoft.SqlServer.Smo)

If I perform a backup with CHECKSUM enabled in the backup options, then does it look like the restore will work on the other server or is the database itself corrupt?
Check the file permissions on the backup on the second server.  It's possible that's your problem, and it's easily fixed.  

If that doesn't fix it, you should run a dbcc checkdb on the original database, which will tell you if the database itself is corrupt.  

If DBCC returns okay, run a RESTORE VERIFYONLY FROM DISK = 'path to backup' on the backup on the original server.  

If all of that returns okay, go back to the second server and double-check the file permissions.