Do web servers reject certificates that do not have an extended key usage with TLS WWW Client Authentication set (if client authentication is required)? Similarily, do Browsers reject certificates that do not have an extended key usage with TLS WWW Server Authentication set? The Extended Key Usage is often flagged as non-critical so theoretically implementations MAY ignore it but the question is whether most implementations require an extended key usage to be present?
Also... where exactly is specified what these two key usages mean and what implementations must do with it? https://tools.ietf.org/html/rfc5280#page-44 does not really specify the meaning behind these key usages (except for what the name TLS WWW Client/Server Authentication already tells you).