Plain flour is for general baking and cooking. It can be white, brown or wholemeal. It is an all-purpose flour so can be used for pastry, batters, pancakes, biscuits and thickening sauces.
Self Raising flour can be white, brown or wholemeal flour, and has baking powder added. Self Raising flour is used for baking where a cake needs to rise – for example in recipes for scones, muffins and cakes.
No, it is not necessary although some recipes recommend doing so in case flour has settled whilst being stored and transported, or to remove any impurities.