Hello executer of the impossible ;)

How did you get the Beckhoff PLC running with the RF680R reader via RS422?

The latest FW for the reader is V4.2. It can be found here.

The status bar in the web interface shows if both connections are established (RS422 and OPC)

There's no test application for a PC to test RS422 as far as I know. You could only try tools like teraterm, putty or hterm. But you'll need to know what to send to the reader (you should be able to find that in the code running on the PLC).