I have definitely run into trouble with 5.0.51a, but nothing very critical. The only issue I remember is a problem with sorting in Wayfinder. OTHO, I was running a *much* older version of MODX then, and I wouldn't be surprised if newer versions take advantage of the many new features in later versions of MySQL.
I can't imagine why they don't want to upgrade to a newer version. The version they are using is extremely out-of-date. MySQL 5.1 came out in 2005!!! (which makes it about a hundred years old in computer years). The MySQL site doesn't even list 5.0 versions any more. The current version is 5.5.
http://dev.mysql.com/downloads/mysql/5.1.html
Also, the fact that they haven't upgraded MySQL for at least 7 years says a lot about their quality as a host. Personally, I wouldn't even consider them as a host, even if they paid me to host there.
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PLEASE, PLEASE specify the version of MODX you are using.
MODX info for everyone:
http://bobsguides.com/modx.html