Not only the right size socket for your lug nuts, but a wrench or bar large enough to actually break the nuts loose.
absolutely. Get a "four way" + shaped lug wrench and a piece of pipe to fit over it approx 18-24 inches long.
They're called a cheater bar - no self respecting Bubba would admit to needing one, but no Bubba would be caught without one.
Then find somebody to show you how to change a trailer tire and actually PRACTICE IT at home. The side of an interstate at night is not the time to figure out how to do it. For the sake of you and your horses, don't expect to wait for some nice guy to stop, his nickname may be "Slash".
Beyond a standard flashlight, also get a strap on headlight to goes on your head. They are like a third hand. Great for flats etc, but beware they may spook your horse.
Some great suggestions! Whenever I go away from the farm riding, I wear a Road ID leftover from cycing days. It has vital contact info on it. I have most of my xtra tack in my trailer just in case anything should break.