Hi dt,
for your first question, the issue with the REX losing connection and reconnecting again every five minutes or so: the answer is no, that is not how it should work. Losing the connection is a bad thing — even if it picks up again automatically — not only because it will affect the eye tracking reliability but also because the user calibration will be lost. Chances are that this issue is connected to the second issue though, so please see if it persists after solving #2 and contact support if that is the case.
Second, judging from your lsusb output, the device is identified correctly as a virtual ethernet interface, but it might not be fully configured yet — it can differ between Linux distros and versions.
You can check whether the eye tracker has been mounted as a network interface by issuing the command ifconfig -a. There should be an interface called usb0 or similar, and if you are lucky then there will also be a line starting with inet addr that shows the IP address bound to the interface. If there is no such line, use the command sudo dhclient usb0 to assign an address and then try the ifconfig again to verify that there is now an address.
At this point it should be possible to run ./fwupgrade --auto --info-only, just like you did, without receiving the 205 error.