The old timers say that you can did horseradish in any month that ends with an "r". I suppose this is done mainly because the crop will resprout in the spring. I typically will cut the top one or two inchs off the stalk and leave it in the ground near where it was dug up. Horseradish is a very tough plant and it's hard to kill once you have it growing. Our patch is only 3 feet square, and it's plenty to supply a couple families.
A couple years ago we ground the horseradish indoors, using a food processor. We quickly learned that the gas released during that process is much worse than cutting onions and would quickly render you into a crying fool. Now we use an antique meat grinder outdoors, which works perfectly. We put the ground material in jars and add a little salt and then cover with white vinigar. They will keep a year or two in the freezer.