Can't login with varnish enabled
The offending parts in your VCL are (line 53):

# Drop any cookies sent to Drupal.

 if (!(req.url ~ "install\.php|update\.php|cron\.php|admin\.php")) {
 unset req.http.cookie;
 }


and (line 81):

# Drop any cookies Drupal tries to send back to the client. This will make it unpossible to login, unless you filter it a bit better that it is made below
sub vcl_fetch {
 if (!(req.url ~ "install\.php|update\.php|cron\.php|admin\.php")) {
  unset beresp.http.set-cookie;
 }
}


Remove both of them.