Unless you plan on the beer sitting in the secondary either really cold or a long time, don't bother adding more yeast. I have let beers sit for over 2 months and they still had plenty of yeast in suspension for carbing.
Whatever sanitizer you use for your equipment will be fine for your corks. Just let them sit in a bowl of the liquid and grab them as needed while corking. Are you using Belgian bottles with the wire cages or wine bottles? In the Belgian bottles, you may have to mess with it to get the right depth but it's really easy after that.