Probably good basic Obedience training would be a good start. And with livestock involved, he MUST do as told, no allowing “sorta, kind of” doing what he is told to do, like Stay. He will quickly decide to make his own choices because things are so exciting, enticing, he thinks he as that option. Around the horse? No, that is dangerous for him and you.
These dogs are crazy smart, both breeds, have a lot of “want to please you” in them, so keep things as fun as possible.
Horse probably should be somewhere near, when you play with the dog and he barks. They do seem to bark often! Horse needs to get over any excitement with that noise, not respond to it. I would NOT let dog run around the horse in pasture or paddock, trying to herd it at all. Dog and horse get excited, dog may get harder in pushing horse with nips so horse reacts by going after dog, running thru a fence.
Dog has to be trained enough to over-ride his chase instinct and STAY when told despite the excitement in front of him. As a puppy, maybe letting him watch from his crate so puppy can’t get himself in trouble if you don’t have anyone to hold and silence him during horse lessons.
Have you trained Obedience dogs before? Shown them? Getting them well trained is a bit of work, striving for perfect, correcting wrong things RIGHT NOW, but not punishing either. Praise is a huge factor in dog having a good time, some treats maybe for really hard stuff.
My friend has a Heeler, great Obedience dog. But the dog needs a job most of the time to prevent it “being creative” in making herself a job. Friend has trained dog to pick up yard trash that blows in, put it in her basket. They walk and pick up road trash often. Dog will fetch sticks that fall down, do a series of tricks. All work to keep dog busy, helpful and not destructive.
ADS rules say no tying the dog to the carriage, EVER. Depending on size of dog, style of vehicle, dog may sit beside you, on the floor at your feet, many sit in the back in modern carriages because steps are easy to get on with. I would make sure dog has something grippy for under his feet, no slipping on metal floors or plastic seats in rougher going or with some speed. Our big dog likes sitting in front on the floor, but she can jump up there herself. NOT picking up an 80# dog!
Classes with your puppy will help socialization, letting him learn to pay attention to YOU wherever you are with him. Trainers usually have several ways to work with a dog and owner who are not communicating well at certain points in training. Don’t feel like all the other dogs are doing better, they just have DIFFERENT issues with their training. Ha Ha.
My dogs are not little robots to be show ring perfect, but they are obedient. They sit and stay when told to, here at home or in crowds of thousands. They stay with us in a heel position, slack leash, not lunging off on the leash, I hate being pulled by a dog! They are civil, friendly, do as requested, come when called despite any distractions. They don’t chase horses or other things when told to lay down and stay, don’t yap when told to be quiet. They are nice dog citizens to have around, take places.
THIS TOO can be your dog, with a little work! Ha Ha 10-15 minutes a day is enough to work your dog, gain on his skills towards being well trained. Just has to be pretty much daily, keeping it enjoyable for both of you. Best of luck with him, probably will be darn cute with that cross of breeds. Keep him busy, he will be very happy with a job.