How to prevent non-English characters from turning into jumbled words?
This is a bit of a guess, but it sounds like some part of the process isn't UTF-8 clean.  If Drupal is displaying everything properly, then it sounds like the database actually is UTF-8.  This leaves phpMyAdmin.

I would log into phpMyAdmin with Firefox.  Right-click, and do View Page Info.  Make sure the encoding say UTF-8.

Then install the Live HTTP Headers plugin.  Download a backup and restore one with the plugin running.  Make sure the Content-Encoding is always UTF-8.

If any of them aren't, then you have some server-specific configuration to fix.

Personally, I think phpMyAdmin is a terrible tool and a security risk.  I use the legacy MySQL Administration Tool for most of my server work and a fair number of backup tasks.  You can also try the Backup and Migrate module for taking and restoring backups.  You can also use the sql-* and/or bam-* commands in drush.