Setting up a PC client on an Ion

The software as such is for free, it's called ETCnomad Eos-Family. You will need a computer with touchscreen to run the software on and network infrastructure to connect it to the Ion (you need to use cable, WiFi is not supported). And last but not least you will need an ETCnomad dongle. The smallest one available should be enough for what you want to do, so that would be an ETCnomad dongle 256. You should ask your ETC dealer for a price.

I thought the dongle is only required if a PC is running Nomad and operating in Primary (acting as the main console) or Backup mode. Am I wrong in thinking that Client and Mirror mode don't require any dongle as they're merely means of remotely viewing or controlling the connected console?

you're correct that mirroring doesn't need a dongle. but if you want a separate device to control the system with its own command line (i.e. a client), you will need a dongle.

I never realised that. I assumed that merely connecting up the primary console to a remote PC with Nomad running would enable client functionality. I take it that the channel count is irrelevant and the criterion is simply the presence of any Nomad dongle? I have a 512 channel output dongle, so I could use this in conjunction with an ETC Ion which has 1024 outputs enabled, couldn't I?

Yes you could. For a dongle the output count is irrelevant if the machine with the dongle is running as client

On that note, if I want to connect the console and client PC together, without needing any other additional devices on the network, can I simply use a crossover Ethernet cable between the two and enable DHCP on the Ion?

You can, but i would recommend to use a switch and static IPs because else the startup sequence and its timing matters. With the switch and a static IP there's less that can go wrong

Unfortunately, I don't have any network switches spare. Will a crossover cable with each device assigned a static IP address not suffice?

It still requires both devices to be switched on not too​ long from each other. If either starts and doesn't find anything on the other end of the network cable it will switch off its network interface. Unfortunately this time depends on the manufacturer of the network interface and is rarely documented.
It can work well for a while and one day you go for a smoke between starting the PC and the Ion and then it doesn't. And i will not work again until you shut both devices down and start them again close to each other.

It still requires both devices to be switched on not too​ long from each other. If either starts and doesn't find anything on the other end of the network cable it will switch off its network interface. Unfortunately this time depends on the manufacturer of the network interface and is rarely documented.
It can work well for a while and one day you go for a smoke between starting the PC and the Ion and then it doesn't. And i will not work again until you shut both devices down and start them again close to each other.