Thanks, I did manage to get there on my own which means I must be learning something. It's great to hear your reply matches what I figured out!
So,
For future reference, to use a USB-ethernet adapter with piCore 9 on a pi zero:
Download
http://repo.tinycorelinux.net/9.x/armv6/tcz/net-usb-4.9.22-piCore.tczPlace it on the SD card: /mnt/mmcblk0p2/tce/optional/
Add "net-usb-4.9.22-piCore.tcz" to /mnt/mmcblk0p2/tce/onboot.lst
1. I think I did this last step, but how can I check? What is the text editor in TinyCore?
2. I did sudo reboot and have ethernet (ifcongig shows eth0), so that's great.
3. I did tce-load -w TC.ycz and it downloaded everything (just like the youtube video I mentioned above)
4. I then celebrated with a cheer (prematurely at it turns out) then typed startx and get
-sh: startx: not found
After 7 hours, I feel like I'm so close!
Once I've got this working I'll write a full How-To for installing PiCore on a Pi Zero for newbies like me! Should help open up TinyCore to more of the younger Pi generation!
Thanks in advance for help with the (hopefully) final step...