Open bridle versus noseband: One good reason I heard to use a noseband in high-impact or high-risk situations is because the noseband reduces the potential gape of the jaw and therefore lowers the risk of a broken lower jaw if the horse falls on its face. The lower jaw is relatively fragile. Breaking it is not fun and involves expensive surgery, or euthanasia. A decade ago my parents had a stallion who broke his jaw when a young colt got into his enclosure when the electric fence had an (as yet undetected) fault. The jaw was pinned because the horse was much loved and there was a vet in the family. Broken jaws aren't all that common, but some showjumping associations at least used to prescribe nosebands for that reason. Educating to cart, your horse can end potentially up in a tangled heap if something scares it, so I'd use a noseband at least until the horse is less green.
Blinkers / blinders: Opinions will vary, but my father has been a harness reinsperson for thirty years and never uses these when educating a young horse to cart. He prefers to make the horse very familiar with the cart and equipment - letting it sniff all the gear, rubbing the gear on the horse, etc, until it's not bothered. Then he drops the shafts over their back from the side at first, both sides, and when they're happy with that he drops them into the shafts for a few seconds, lifts them again, increases the time. When the cart first goes on, it's only placed in the loops, not tied in, so it can be quickly disconnected if the horse startles, and there's always a "babysitter" walking at the horse's shoulder, and the reinsperson long-reining from the back. Sitting in the cart happens progressively once the horse is good to hitch to the cart with straps and all. At no time does my father hide anything from the horses, he just progressively familiarises them with everything until they're happy. All this goes on in a safe environment away from traffic and fences and hoo-haa. Blinkers have only been used a few times in all these years as a temporary educational measure to stop harness horses from jumping shadows at night-time track meetings.
Age: The earlier you familiarise a horse with equipment, the better. This is not the same as working a horse hard, which needs to be avoided until the horse matures - and conditioning for work is a gradual process. Horses can be taught to accept a cart as yearlings, and have a saddle strapped on them as yearlings, although the rider doesn't happen till later! If something is "old hat" to a horse, it will be so much calmer.
Your context isn't harness racing, and you may have "scarier" carts and buggies, and want to drive horses as teams. But quite a bit of the underlying groundwork is the same.
Oh, and doing courses and witches' hats and strange surfaces etc when long-reining is an excellent idea: Have your horse familiar with all that BEFORE you hitch it to a cart. It much reduces the risks of bad experiences.