Macs using MacTCP or Open Transport with the default settings do not load the TCP/IP protocol on startup. TCP/IP is only loaded into memory when an application that uses TCP/IP is run. The Mac will only respond to a ping request if you have already run an application that uses TCP/IP.
That might be one of the strangest things I've heard. So, it looks like TCP/IP might be working, I just have to be running a program that uses it.

Well, I can tell you now that the ethernet was working all along. I successfully loaded the NCSA Telnet program to the 7200/120. Once it was open, the Mac responded to ping.
I also put OS 9 on the 7300/180. It's good to have a (slightly more) modern OS that can get on the web, burn CDs and format floppies. In case anyone was wondering, OS X cannot write to or format a floppy. At least not any version within the past few years...
The only thing I have left to do is make an image of the currently-working setup. It turns out that Acronis won't do it and something like Carbon Copy Cloner in OS X won't either due to the older filesystem used in System 7.x. It's possible I can get something going on the OS 9 Mac, but I think my best bet is simply to use 'dd' in Linux.
I had to buy a PCI SCSI card to make this attempt. It's a shame I probably threw away at least half a dozen of them in the past few years.
I'll let you know how that works out once I cool off.