Update on Cider Vinegar:
One batch has done nicely and now has a healthy mother.
As for my suspition of freezing a "starter", when I racked (actually I filtered as I'm not worried about clarity) I froze the solids. A week ago, I started a new batch and broke off the smallest chunk to drop in the half-gallon of apple cider. By the next day, it was bubbling like mad. So it does work for what we're trying to do.
Now I need to freeze some "mother of vinegar" from my first batch. I've also read that you can dehydrate a mother, so maybe I'll try that, too.