1) I use mineral oil. You can find it in the pharmacy section as a laxative.
2a) Does yours look like this?
Undo the 2 screws on top and remove the cover. Lightly lubricate the now-exposed springs and boxes. Also lube the hinge at the base of the handle. Wipe off any excess. DO NOT test the corker with the lid off - the springs could fly upwards.
2b) Most of the parts are already exposed though. So you might be able to lubricate with the lid on. Get the hinge, and the insides of those brown boxes that squeeze the corks. Then lower the handle a few times to work the oil in.
3) For sanitizing, I put a little on a paper towel and run it around the parts that'll touch the cork or bottle. Then dry off when I'm done corking.