Did you wash the horse before you tried to clip him? Dirt will dull blades and make clippers virtually useless in a matter of moments. That is a WHOLE other topic that has been covered plenty on the main mini forum if you do a search.
Well, the reason the cart is sliding forward is because you don't have your breeching straps through the holdbacks (the little rings on your shafts). That is your slowing/braking system with harnesses that have open tugs (aka "shaft loops"). Without it, you risk the cart slamming into the back of the horse, which you are figuring out.
There are a number of "correct" ways to "do up" your breeching straps, but generally you put the strap in the ring from the inside towards the outside under the trace, wrap the strap around the shaft in front of the ring as many times as necessary to take up the slack (more than three means you need to move your ring). Then put the strap under all or the first wrap and buckle it back to itself, over the trace thus catching the trace in the "loop".
The absolute best way to get your harness adjusted is to have someone knowledgable look at it. This DOES NOT have to be a mini driver/trainer, as it is the same for big horse drivers as well. I would not plan on entering a show until you've had at least one lesson with someone. Driving is not as simple as basic riding, as there is a lot more to it. There are some good resources on my website for beginning drivers, and I would also invest in some good videos and books if you can't find an instructor in your area (also listed on my website).
I think your horse is in good "rig" and really doesn't need a neck sweat, stable sheet, or any other contraption. We don't use any of that and have had champion driving horses for a number of years. Good quality work, along with adequate turnout and excellent feed creates good driving horses. Neck sweats are a fad for halter horses.
Myrna