Not only is James (yesterday's cat in the tunnel) home safely, but he was whisked off to the vet today. Turns out he has Horner's Syndrome, which is causing one of his third eyelids to stick up. This will go away on its own, the vet said. No spinal problems, or ear problems. He does have a fever (103, which isn't *too* high for a cat, but is elevated for an adult), so he was put on a week's worth of broad spectrum antibiotics. He's being tested for feline leukemia, but the results won't be back until tomorrow. (And thankfully the neighbor knows enough to know that FeLeuk isn't an instant death sentence, even if it comes back positive, which it hopefully won't.) He's eating and drinking and seems to be moving okay now, so while he's not entirely healthy, he's doing pretty well.
I told her I'd keep my fingers crossed for him to have a negative result on the test and to keep getting healthier.