XML document parser with XPath
Based on the naming guidelines properties shouldn't be named using some kind of Snake_Case casing.  They should be named using PascalCase casing.  



Without seeing OutboxResponse's  implementation and its other usages, I would suggest initializing the Violations property in the class itself.  



The loops inside the switch cases are very similiar. They only differ by the used "prefix" e and r. These should be extracted to a separate method to reduce code duplication.  



You have a lot of vertical space wasted by using to many new lines. This reduces readability.  



English isn't my first language so this could be wrong, but I guess "Errors was found in packet" should be "Errors were found in packet".  



These variables  


string violationCode = "g:body/g:code/text()";
string violationLevel = "g:body/g:level/text()";
string violationName = "g:body/g:name/text()";

string itemResultResult = "g:body/g:itemResults/g:itemResult/g:result/text()";  



aren't used and should be removed.  



You should always declare variables as near as possible to their usage.  



By extracting the creation of the XMLDocument to a separate method you can reduce the size of the method which adds readability.  



If you need a comment to describe for what a variable stands then this variable isn't  properly named.  



After applying the mentioned points the refactored method we would get  

public OutboxResponse CheckSendResponseForOutboxPacket(Outbox outbox)
{
    XmlDocument responseDocument = GetXMLDocument(outbox);

    XmlNamespaceManager nsmanager = new XmlNamespaceManager(responseDocument.NameTable);
    nsmanager.AddNamespace("g", "http://google.com");

    var rootElement = responseDocument.DocumentElement;

    OutboxResponse outboxResponse = new OutboxResponse();
    outboxResponse.Violations = new List&lt;OutboxResponseViolation&gt;();

    string prefix;
    string xpath;
    string resultPath = "r:body/r:result/text()";

    switch (rootElement.SelectSingleNode(resultPath, nsmanager).Value)
    {
        case "success":
            outboxResponse.Result = "Packet received without errors";
            xpath = "g:body/g:itemResults/g:itemResult/g:violations/g:violation";
            prefix = "g";
            break;

        case "failure":
            outboxResponse.Result = "Errors were found in packet";
            xpath = "g:body/g:violations/g:violation";
            prefix = "r";
            break;

        default:
            return outboxResponse;

    }

    XmlNodeList nodes = rootElement.SelectNodes(xpath, nsmanager);
    outboxResponse.Violations.AddRange(GetOutputViolations(nodes, nsmanager, prefix));

    return outboxResponse;

}
private XmlDocument GetXMLDocument(Outbox outbox)
{
    var content = outbox.Outbox_content.FirstOrDefault().response_content;
    XmlDocument document = new XmlDocument();
    document.LoadXml(content);
    return document;
}
private IList&lt;OutboxResponseViolation&gt; GetOutputViolations(XmlNodeList nodes, XmlNamespaceManager nsmanager, string prefix)
{
    IList&lt;OutboxResponseViolation&gt; violations = new List&lt;OutboxResponseViolation&gt;();
    string levelPath = prefix + ":level/text()";
    string namePath = prefix + ":name/text()";
    string descriptionPath = prefix + ":description/text()";

    foreach (XmlNode node in nodes)
    {
        violations.Add(new OutboxResponseViolation
        {
            Level = node.SelectSingleNode(levelPath, nsmanager).Value,
            Name = node.SelectSingleNode(namePath, nsmanager).Value,
            Description = node.SelectSingleNode(descriptionPath, nsmanager).Value
        });
    }
    return violations;
}