The great computer war may be over.
I tried, as a last-ditch effort, to reformat the Seagate on the Gateway (remember, I reformatted it for the Mac so everything is copasetic on that end -- for now, at least). As expected, that did not work. However, while I was thumbing through the section of the Seagate's instructions I hadn't yet bothered to consult, I saw that Seagate does NOT recommend using their drives to switch back and forth between PC and Mac. So, yeah, this ordeal was pretty much my fault. Whoops. That's what I get for cutting corners.
So anyway, I bought a 1GB flash drive and transferred most of the crucial junk from the Gateway to the Seagate and/or the Mac. I rationalized that expense as being probably what I'd pay for a tech geek to fix it. Of course, the printer still doesn't work, but then, it didn't ever really work with the Gateway after we had its hard drive replaced the first time.