If you look at the rubber bumper that slides over the end of wrap handle, it is slid up very high. I have always installed that bumper low enough that it will engage with the case when the saw is pinched, and will limit the amount of flex between the tank handle and the rest of the saw. Where it is sitting on your saw, it is swinging in the breeze and it sounds like you have too much flex that is causing your throttle arm to slip off the carb. Also, your mounts may be worn or broken.
As for the starter rope length, I always do what Stump Shot does - wrap on as much rope as the pulley will take without over filling the pulley. A slightly smaller diameter rope will give you slightly more length but will be weaker. Too small and it might break, or cause the pulley sides to start to spread and the rope will bind. If you remove the pulley to install the new rope, clean everything well (including the spring). You will be amazed at the crap packed into there. Then lightly lubricate the spring and post. I use sewing machine oil on the spring and a slight amount of grease on the post. Your starter will be silky smooth when you are finished.