The arp command you would use to find out who is connected.
Wrong. It shows the ARP cache of this host. If there wasn't a reason to resolve the IP to a MAC address the table will not show an entry although the device might be connected.
In the meantime I did some tests (also in windows) is not until I discovered that the server is pinged, Arp does not show its ip
The reason for this is found in the above answer.
What do you have from the server if you don't know the IP address? Is only the IP changing or is the MAC address changing too? The former case can be fixed by setting a fixed IP for the MAC address in the router's DHCP configuration.