As @bmike said, Internet Sharing hides a lot of complexity behind a very simple interface, and some of your questions can't be answered authoritatively without interviewing some of the Apple engineers behind it. But that won't stop me from taking a stab at it...
1) AirPort is different from the other interface types because in order to share over AirPort, your Mac has to actually create the wireless network (as opposed to just providing service over an existing ethernet, FireWire, etc connection). This means that InternetSharing needs to have a bunch of info about how to create the wireless network: network name (SSID), channel, security, etc.
2) Resharing over the same ethernet interface is useful under some circumstances. For example, on my home network my ISP provides limited number of static IP addresses for my use. I run a Mac doing the equivalent of Internet Sharing (actually, I set up the daemons manually as @Spiff recommended) to reshare over the same ethernet. Result: if I put a computer on my home ethernet and config it via DHCP, I get a private (behind-the-firewall) IP address from my virtually unlimited internal pool. If I manually config the computer with one of the public IPs, I get full unfitered internet access, but use up one of my limited public IP pool. Because they're both on the same network, "moving" a computer behind or in front of the firewall is just a simple configuration switch.
On the other hand, if you did this same trick on an ethernet network that already had a DHCP server, computers attaching to the network would randomly get configuration from one server or the other, leading to unpredictability, confusion, and hair-pulling. It's definitely a use-only-if-you-know-what-you're-doing feature. Fortunately, Internet Sharing is smart: if it detects another DHCP server on "its" private network, it shuts itself off to avoid trouble.
3) I don't know of a way to change the private IP range on an IS-created wireless network. On the other hand, it shouldn't really matter, since the network is being created by Internet Sharing, and therefore it doesn't have to worry about conflicts with any existing network numbering.
4) You can add interfaces with Apple's USB Ethernet Adapter. Get some USB hubs, and pile them on!
Disable IPv6 and WiFi. Plug both in via Ethernet. Verify your targets IP. Turn AFP and all other sharing settings off on the target, and wait for it to disappear from the sidebar on the machine you're connecting from. Then re-enable AFP sharing on the target, and see it come back on the source.
Ping the IP, or even the name (foo.local if the computer's name is foo). Try connecting by clicking on it from the sidebar - failing that try using the Go menu (connect to server) and putting in a URL in the format afp://<name or IP here>.
Let us know how it goes.
EDIT: then, once that works properly, change one thing at a time. Enable WiFi on one machine and remove the ethernet cable. Then test. Then do the same on the other machin, and test. Then enable IPv6 on one.. test. Then the other. Etc, until you've worked your way back to how you want your machines to be set up. This can also help us isolate where the problem is happening (if you re-enable WiFi and sharing stops working...)
Best Answer
I fixed it! What wound up working for me was going to the Network prefpane, deleting the Ethernet and Wi-Fi interfaces, and re-creating them.