The great thing about driving people is that usually they are very willing to share information. If you can't afford an instructor, then at least find a local driving club where you can meet other drivers who may be able to help and mentor you. Believe it or not, even big horse drivers can help a new mini driver.
If you make connections with other drivers, you might be able to get the opportunity to drive a horse before you try to train one to drive. Our club has trail drives and sometimes members with horses that aren't ready for the trails will just bring themselves and catch a ride in a vehicle with another driver. That is GREAT experience. Would you break a horse to ride if you didn't know how to ride? The challenge and difference with learning to drive the hard way is that if you get bucked off a riding horse, you can get back on. If you have a wreck with a green driving horse, he may never drive again. It's too traumatic to have that thing chasing him. (We ruined our first horse that way, taking him in a parade in which he wasn't ready for. He never drove again...and we tried.
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You aren't missing any singletree hooks. That is somebody's version of a sword end singletree, and a scary one at that. The trace slots fit over the ends, and like has been said, something holds the trace on by being fitted through that hole, like a bolt. I would replace the entire "singletree" with a real wood one, either sword end or hook end. Or you could get some snap shackles for that hole, but they are about $20 a piece. Don't cheapen out and get any old snap. That is a point of draft and failure, and you need a strong connection.
And then the singletree needs to be fitted with straps to limit its own swing. They should have just enough play as to not interfere with the movement of the horse's shoulders. The purpose for the straps is to keep the singletree from smacking the horse in the butt if one of the traces comes off. (You can read a story about what happened to somebody who didn't put on those straps on my website on the Driving Safety page.) It gives you just a bit of time to stop the horse before something worse happens. (Go read the story...
http://rhinestone-ridge.wikispaces.com/Driving+Safety.) Even simple dog collars do the trick, as long as they are in the right spot.
There are also a couple of "He's just a little mini. How much harm can he do?" stories, too, on my website, which is why people here are trying to convince you to get some help. The more you learn, the more you realize the less you know. I have clients that can't believe some of the things they did when they were first learning to drive and didn't know any better now that they know more. They said they had angels on their shoulders, because they could have had some awful wrecks. Regardless...
Your cart is definitely too big. Maybe you can have the shafts cut down and bent in some. That is what my mom did with our first cart. I would also make sure those are true cart wheels and not bicycle wheels. Bicycle wheels are not made for the sideways torque a cart puts on them.
It's good that you are getting this input at this time in your process, but another "mistake" you've made is to have the horse in the cart without a bridle and tied to the fence. I understand what you are trying to do, but this is not accepted practice for driving horses. There is SOOOOO much to learn when starting to drive.
If it helps, there is also a list of resources (books, videos) on my website on the Links page, as well as articles I have written on the Cart and Harness Education page. When I have time in the next few weeks, I hope to add an article focusing on cart parts, fit, and care. But before you go out and purchase a book, I would take that $$ and apply it towards even one lesson on a finished driving horse, no matter what the size. That will be money well spent.
Myrna