Yes, but depending on the quality and age of your softner you may have a couple side-effects. Your water may taste salty, and/or you may lose some water pressure.
Yes, you can run the water while the water softener is cycling but it will delay the process. By taking out some of the water that has been softened, new water will come in and that will need to be softened.
A water softener uses a medium that serves to exchange "ions" of calcium and magnesium with sodium and potassium. Hard water contains large amounts of calcium and magnesium. Soft water using meets all health regulations and is known to be safe.
