Chrysanthemums are usually propagated from cuttings. These are new growths that appear on the old plant during winter and early spring. When cuttings reach about 2-3 inches in length they can be snapped of and propagated individually to form new plants. The 'Getting Started' page has some pictures and more information on propagation. In some areas it may be a little later as there is still a risk of frost that may damage the tender young growing tips.
Most growers use some form of control against aphids, blackfly and earwigs.
As blooms begin to form they present an ideal place for such pests to hide. Because of this many growers resort to the use of paper bags that are inflated and placed over the developing flower, thus keeping these pests at bay. The 'Bloom Protection' page shows several methods for bagging blooms. Yes chrysanthemums do grow from seeds but it's likely that every plant grown from seed will be different both from the plant next to it and from it's parents. Many new cultivars are created this way.