Most of my lucid dreams start as false awakenings as well, as for reason why we have lucid dreams often start like that, is that i think we have much higher awareness in dream during false awakenings and when we practice lucid dreaming we learn to easily turn them into lucid dreams, thus if we have many of them, we'll get lucid from the most of time.
As for dreams lasting longer, you might want to look into stabilization techniques to stabilize the lucid dream when it starts or when it starts fading out, here's tutorial: Dream Stabilization and Clarity Tutorial - Lucid Dreaming - Dream Views .