Is copying the /var/lib/mysql directory a good alternative to mysqldump?

This approach is only going to work safely if you shut the database down first. Otherwise you could well end up in an inconsistent state afterwards. Use the /etc/init.d/mysql stop command first. You can then restart it after the backup is taken.

• Yes
• Yes if the table is using the MyISAM (default) engine. Not if it's using InnoDB.
• Probably not, and if there is, you just need to execute mysql_upgrade to fix them

To avoid getting databases in a inconsistent state, you can either shutdown MySQL or use LOCK TABLES and then FLUSH TABLES before the backup. The second solution is a little better because the MySQL server will remain available during the backup (albeit read only).