You can use the HTTP_COOKIE server variable from the Request object to get the cookie string that was included in the request.
string cookieString = Request.ServerVariables["HTTP_COOKIE"]
If you want to get the session cookie name from web.config, why don't you add a simple entry in the appSettings section containing the session cookie name
<appSettings>
<add key="SessionCookieName" value="__SessionCookieName"/>
<appSetting>
<sessionState cookieName="__SessionCookieName" />
web.config, :
public static bool SessionCookieName
{
get { return ConfigurationManager.AppSettings["SessionCookieName"]; }
}