It is likely a flat plug with a hex socket hole. Scrape the dirt and grass off near the oil fill area. To drain the oil I usually top the mower up 90 degrees into a pan.
You mean by the throttle control being all the way past fast? The choke is usually operated by a wire coming off the speed control arm that only activates when past full throttle.
You may have to get a different control plate and the right activator wire.
Perhaps the camshaft is bent from when the rod broke. I put it in the base hole and turn it to see if it wobbles. And make sure the base hole is not cracked.