Zen Cart has no problems running with MySQL5. Indeed it's getting increasingly rare to see a server running an older version.
The error message gives a clue as to the problem. It appears to be passing a path name rather than a table name to the mySQL engine.
Normally I would have said that this was a problem with your configuration files. I'd certainly have a look at includes/configure.php and in particular the DB_PREFIX setting towards the bottom. Normally this would be empty (i.e. two apostrophes) or something like 'zen_'.
However, it's interesting that you say that you could preview the site through Plesk until you changed the nameservers. That would point to something hinky in your server setup and you may need the help of the people hosting your server to get to where that directory path, "./dashequestrian/" is coming from.




