Thanks for your reply - since I've asked the question, I though I should post a partial answer.
I don't know what was the exact cause of those error messages - I would guess that apache process was overloaded.
The following configuration changes seemed to eliminate those error messages on our server:
ExtendedStatus Off (in apache2 config)
commenting out sections
# WebworkSOAP "bridge2" handler
########## WebworkWebservice SOAP handler ##########
########### WebworkWebservice XMLRPC handler ##########
# WebworkSOAP WSDL HANDLER :: TO BE REPLACED WITH A FILE FOR PRODUCTION SERVERS
# WebworkSOAP handlers (for integration with moodle)
########## WebworkWebservice SOAP handler ##########
########### WebworkWebservice XMLRPC handler ##########
# WebworkSOAP WSDL HANDLER :: TO BE REPLACED WITH A FILE FOR PRODUCTION SERVERS
# WebworkSOAP handlers (for integration with moodle)
in webwork.apache2-config file. I'm not sure why those sections were by default active in webwork distribution we checked out.
We probably should use lighthttpd to serve static content, and let apache2 do the heavy perl work only.