As long as they dont give off an odor (rotten potatoes really stink!), or have green spots & roots growing from the nodes, they're perfectly edible. peel & place in a pot of cold water to crisp them somewhat & use.

Great for making hash with other leftovers.

This recipe is actually intended for "old" potatoes! http://www.food52.com/recipes/2532_tads_roasted_potatoes

You can still use them, just as long as there is no mold growing on them, be sure to remove any green parts too. It will taste okay and even firm up once cooked, especially if your making fries.

Not necessarily, but they will not taste or have as nice a texture. Make something like garlic mash to hide softness