Please read page 238, When demand curve intersect, the flatter curve is more elastic,
The elasticity is not the same thing as slope. While slope is calculated using changes in quantity and price, elasticity is calculated using percentage changes. But it is true that if two demand curves intersect, the one with the smaller slope (in absolute value) - the flatter demand curve - is more elastic, and the one with the larger slope (in absolute value) - the steeper demand curve - is less elastic. In Figure 6.1, for a given change in price, demand curve D1 is more elastic than demand curve D2.

Yes, along one demand cuver, the price elasticity of demand is keeping changing, because Elascity is calculated by using percentage change not absolute value.