Is there such a thing as 32-bit Windows 8.1?
There is, but you really don't want it. You'd end up seeing maybe 3/8ths of your total RAM in your new computer. At this point mobile phones are making the switch to 64 bit, no need to go backwards on the desktop side.
Or a compatibility mode that runs 32-bit on the 64-bit machine?
Windows has a compatibility mode for 32 bit applications (odds are many programs you run are making use of this), but not drivers. Being that the issue is a driver issue, unfortunately no help here.
I guess I am confused about whether 32/64 bit is a hardware thing or a software thing. It was my impression that it is strictly hardware, so I have no possible USB option with this computer.
It's both. Your old system was actually 64 bit too, but you had a 32 bit version of Windows running on it. The hardware (mostly the BIOS/EFI, CPU and RAM) have to support 64 bit addressing. Then an OS also has to be 64 bit to make use of anything beyond the 32bit barrier. And finally a program also has to be compiled for 64 bit to make use of extra memory, or the new CPU instructions introduced.
So, apparently I need at least one more piece of hardware. That's OK, I am on good terms with my local computer store. Would that be a hub, or a switch, or another router, or ??
You would want a switch. You don't want another router, and hubs are pretty much impossible to find these days.
My advice would be to just unplug the printer from the network, and plug that into the empeg in the rare times you need it. If you find you are doing this more then a few times a year, then perhaps it's worth adding the switch to gain another ethernet port on your home network.