You can do it like this:

var myvaluetoextract = htmlDocument.DocumentNode.SelectSingleNode("//div[@class='paste-copy-url']//input");
var value = myvaluetoextract.GetAttributeValue("value", null);

//input means you search for input elements in the div's subtree, recursively. GetAttributeValue is a helper that will never fail, even if the attribute doesn't exists (in this case if will return the 2nd passed parameter - which is null here)