For Corkscrew you park beside the paved highway on the way to Beaty. Surprise Canyon is in the Panamints, just north of Ballarat. It's good gravel road, if sometimes a bit wash-boardy. All quite doable with a 2wd rental.
Bodie's a great place to explore, but I don't know if you can get in there during the winter. It's a good road, but not plowed.
