This is less a recipe than a method and is open to serious variations. Some recipes call for a 70/30 ratio of salt to sugar. Spices can also be added (clove and/or cinnamon, for example) to bring other flavors to the party.
You can preserve other citrus this way. Keep in mind, though that fruits like oranges aren't quite as acidic and will need to be reinforced with lemon juice.
Many recipes call for the lemons to be "nearly" quartered, by not cutting all the way through the lemon so that the four pieces remain connected to each other at one end. I'm not sure if this is just a traditional hold over, or a it's intended to make it easier to dole out one lemon's worth of preserves. Whatever the case, I quarter mine completely so that they'll fit better into the jars. I've had no issue doing it this way, and quarters seem to me to be a good measure of to work from when adding this ingredient to my own recipes.
Some recipes call for the jarred lemons to be transferred to the fridge immediately. This method was invented well before refrigeration even existed, so I'm not sure it's really necessary. I can't imagine anything could grow in such a salty acidic environment. I tend to leave mine out on the counter indefinitely. That's up to you. Your mileage may vary. Buyer beware.
A pickle stick (or kraut stick) is used to press cabbage or other vegetables into a crock for fermentation.