A lucid dream is a dream in which you know you are dreaming. If you knew that everything was a dream, during the dream, then you were lucid.

Seems like you have awesome dream recall. Recording your dreams is a great way to start on your journey to lucidity. It also serves an obvious purpose for dream continuation (how can you continue a dream, unless you remember it? How do you know you successfully continued a dream, unless you remember it?). Our forum provides a place to post your dreams: Dream Journal.

Continuing the storyline is the easiest. Probably easier if you know you're dreaming. All you have to do is keep the storyline in mind. If you believe it is continuous, then all your dream characters will also believe, and they will behave accordingly. As far as consistency in setting (the mountain), the amount varies, and you may need to practice travelling there in your dream before it's stable.