Thing i like about vespas is that they have a fan to keep things cool, the tin above the cylinder is necessary. If it revs by itself at idle then check your idles speed screw, if the slide is all the way down and you still cant control speed then suspect air leak, Normally the shim between carb n intake is compromised. Also, jetting, no. of filter screens and running an air box or not affects fuel mixture. with carb off clean out that tiny idle jet hole/passage under the throttle slide.
To tighten the slack in the belt you can grind off the rivets in the subframe and move the whole motor forward to take up slack. Starting should be fairly easy using the decompression lever.