Is it possible to give the RevPi Core S an additional network interface to integrate it into a second, physically separated LAN TCP/IP network?
Unfortunately, this will not work. The gateways are fieldbus slave gateways. In your case, it is an Ethernet/IP field bus (https://de.wikipedia.org/wiki/EtherNet/IP). It is not possible to use it as a "normal" network interface. If you need an additional network interface in the system, I would recommend a USB network adapter or the RevPi Connect S. Please have look here: (42) Gateway Ethernet/IP als "normale Netzwerkschnittstelle" einbinden. - Revolution Pi Forum
How can I query the status of the field bus of the other gateway?
Yes, have a look at the user manual there are the following registers available:
Why are there two two RJ45 sockets for the connection
Only if a "Device Level Ring (DLR)", an optional feature of this fieldbus, is used for greater reliability (redundancy), both connections are required.
In addition, the two ports can be used to chain a number of devices without forming a DLR ring. (So you do not need a separate Ethernet switch to operate several devices on one PLC).