I am unable to see anything on the server side that would cause this error with your specific installation. We may need your account specific information to do so. I do see you are working with our support department currently, so they can help you by using your particular account information.
I did find a link in the Wordpress Support Forums that discussed this exact issue. You can view it here at the Serious MySQL Errors post. This may shed some light on the issue for you.
An increased number of visitors should not decrease your Google Page rank, but repeated errors may. We do not know anything specifically about increasing or decreasing Google page ranking with Google, however.
If you have any more questions or information specific to the issue please leave a comment below so we can further assist you.
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