As you might have heard before, the "WILD" technique (wake-initiated lucid dreaming) concentrates on entering lucid dreams right while falling into sleep. So, yes, it's possible to enter an LD directly after falling asleep (although you can't really refer to this state as "being asleep", because during the LD, you keep parts of your conscious mind awake).

Your second question doesn't have anything to do with the time you have been sleeping before the LD, I think. Being aware of having a dream, but unable to control it is called "prelucid dream" and always appears if your subconscious isn't "ready" for experiencing an LD. You can try to use your senses in the dream to get more control, for example, try to take a smell at some objects or concentrate on what you can hear in your dream. In some cases, it helps to improve your lucidity and allows you to get more dream control.