I use a Sophos Firewall and I needed to open some ports for the Ring Cam and Smartphones. Finally I opened following Ports from LAN to WAN:
TCP 5223, 5228, 9998, 15064, 32882-65533
UDP 53, 67, 68, 123, 5353, 5001-65439
Now the live video stream is working and the WiFi Internet Connection Test finishes successfull.