Continuously throwing chains means there's several things that could be wrong: engine mount bolts are loose, engine out of alignment with the rear sprocket, or the clutch and/or sprocket are bent or possibly out of round. Looking at the second picture you uploaded, showing the clutch from the side, it looks to be missing the bolt that holds it on the crank shaft, which the majority of kart and mini bike engines will have. If your engine doesn't use a bolt to hold the clutch in place, it might use a set screw or two, which you'll want to check and make sure they're tight. Next thing I would check is the clutch bushings in the drum, they should be tight with very little play in them and you should not be able to move the drum up/down or twist it except a tiny amount. If the clutch checks out then move to the counter shaft and check the bearings and sprocket, make sure they're tight and don't have up/down or side to side movement. Use a good straight edge such as a square or machinist ruler against the clutch drum and check the chain alignment.
In order to switch to a belt torque converter, you'll first need to pull the clutch off the engine and measure the crank shaft diameter and length. Then take more measurements between the engine and counter shaft, engine and frame, etc. Basically you'll want to make sure there's enough clearance back there to run a torque converter, and get the correct length belt. Max Torque, Comet and several other companies make bolt on torque converters for multiple applications, surely one will be a direct fit for your yard kart. Hopefully you get it figured out.
