Select any element by class - xpath

You're missing a location step after the //, as the predicate on its own is not allowed.

var nodeByClass = doc.DocumentNode.SelectSingleNode("//*[@class='class_name']");
var nodeName = doc.DocumentNode.SelectSingleNode("//*[@name='element_name']");

short-hand for:

var nodeByClass = doc.DocumentNode.SelectSingleNode("//self::node()[@class='class_name']");
var nodeName = doc.DocumentNode.SelectSingleNode("//self::node()[@name='element_name']");