The coil is not the alternator. There is a stater on the engine, behind the flywheel. You're not going to like dealing with it. You basically have to remove the engine just to get all the sheet metal off. Then pull the flywheel. No other way to it. I'd look around REALLY CAREFULLY at all the wiring, connections, grounds, etc., before going to the stater. Unfortunately, youll have to fix it eventually. The PTO switch on the Cub Cadet pulls more juice than anything, and it will kill your battery while you are mowing. Then everything's dead and your stuck until you can charge the battery.
Hydro lever, yes it is adjustable, and it needs to be adjusted. Pull the tin tunnel cover off and the swash plate and adjustments are underneath. The best way to adjust it is to jack the back of the tractor just off the ground, disconnect the brake linkage to each wheel, start the engine and set the brake. Then adjust until the wheels stop turning.
Cub Cadets 682, 1811, 1864, Simplicity Legacy XL 4x4 Diesel with FEL, 60" mower, 50" Tiller