MySQL replication: 'Duplicated entry for PRIMARY key'
ASPECT #1 : Replication
I don't think that
replicate-wild-do-table = db_backup.%
replicate-rewrite-db = db-&gt;db_backup

belong together.
Other people have wondered about this as well

Jun 14, 2012 : Configured MySQL replication but its not working
Jun 13, 2012 : What is the difference between 'replicate-rewrite-db' and 'replicate-do-db ' during mysql replication ?
May 31, 2012 : Mysql replicate-rewrite-db not working

The problem stems from the order replication rules are processed. According to the MySQL Documentation on Replication Rules:

If any --replicate-rewrite-db options were specified, they are applied before the --replicate-* filtering rules are tested.

Even the MySQL Documentation on replicate-rewrite-db says:

The database name translation is done before the --replicate-* rules are tested.

The replicate-wild-do-table is enforced after the rewrite. It would not be surprising if this ordering somehow imposed an INSERT into a table that has data already.
You are probably asking how did the data get there ?
ASPECT #2 : mysqldump
Doing mysqldump --single-transaction would seem to be the greatest way to point-in-time dumps of data. Unfortunately, mysqldump --single-transaction has an Achilles' Heel : ALTER TABLE. If a table is subject to any ALTER TABLE commands, such as a DROP TABLE and CREATE TABLE, that can break the integrity of the transaction the mysqldump was trying to do the dump in. Truncating a table (which is DDL in the MySQL Universe) and dropping and adding indexes can also be as disruptive.
You can find more information on that from MySQL Performance Blog's Best kept MySQLDump Secret. I actually addressed this point in a past question describing 12 commands that can break the integrity of a mysqldump's transaction : MySQL backup InnoDB
CAVEAT

Personally, I would switch binary-format to STATEMENT
Like I said in MySQL backup InnoDB, you should have another slave (without any replication filters) dedicated to doing mysqldumps.
63000+ seconds behind? You should look into doing parallel table dumps : How can I optimize a mysqldump of a large database?

EPILOGUE
One or both of the aspects may have contributed to letting a row slip in during the mysqldump that should not have existed due to either the rewrite rules or the isolation of the mysqldump being overridden.
SUGGESTIONS
I would do a mysqlbinlog dump of all the relay logs since the start of the mysqldump to see all INSERTs that the Slave will process and see if those rows already exist on the Slave. If they do, you could probably do two things:
1 : Skip all the Duplicate Key errors
Simply add this to my.cnf on the Slave
[mysqld]
slave-skip-errors=1062
skip-slave-start

and restart mysql. Then, run START SLAVE;
all the duplicate-key errors will get bypassed. When Seconds_Behind_Master gets to 0, remove those lines and restart mysql.
2 : Download percona tools
The tools you need are

pt-table-checksum
pt-table-sync

Use these to find the differences in the Slave, and then correct them