:welcome:
To give you quality relevant answers we need the model numbers from your mower.
1) the pulley or the pulley are pivot will need to be replaced.
Buy the parts , check them against the originals then put them aside for a few months so you can do it in the off season.
2) Kohler mounts the rectifier on the blower housing , which is insulated from the engine, thus no ground.
They run a ground strap from a blower housing bolt to the regulator mounting screw, which regularly breaks.
I make up a pair of wires to make a ground connection to each of the mounting screws ( belts & braces ).
At full speed the alternator should be putting out 30V AC min at the two outside regulator terminals and 14 V DC from the centre terminal.
3) going to need the make & model of the hydros to answer that.