Both of these problems are caused by a bent crankshaft. When the crankshaft is bent, the engine is thrown out of balance and will vibrate violently. I

Whenever I try to start my mower, the pull cord is violently jerked out of my hand and I can not start the engine.
This problem is most likely caused by a sheared flywheel key. All mower engines use a small square key to index the flywheel on the crankshaft. These keys are designed to shear in half to prevent damage to the engine when the blade or flywheel are stopped suddenly. When the flywheel key is sheared, the ignition timing of the engine is thrown off which will cause it to fire at the wrong time. In these cases, any attempt to start the engine with the pull cord will be met with a violent jerk that can cause personal injury.

There are a number of things that can cause a sheared flywheel key. The most common cause is running over an object in the yard which is hard enough to stop the blade from rotating. In this situation the crankshaft and blade are usually damaged bad enough to require repair or replacement.