1) They do. However, you may just use "бегу вниз по ступенькам", and it would be more natural
2) "Я бегаю вниз" and "Я хожу вниз" are different. Certainly, they do mean that you go/run down. But they are for REPEATED action, like "I often run down there" or "I always go down to the ground floor to get some coffee". Use a one-directional verb for one-time action in a specific direction.
3) "Я съезжаю на лифте" and "Я съезжаю вниз на лифте" are extremely weird, though grammatically correct and somewhat understandable. Don't use it. The verb "съезжать" is never used for elevators. Use it for vehicles going down or going away from some place (from the highway, from the road or from the place the car was standing on).
It is also used for people: when you are moving to a new place (i.e. to live there), you use "переезжать" to describe a process in general, or you may use "съезжать" to say you are moving "away" from the place you are currently living in:
"Мы скоро съезжаем с квартиры на Балаклавском."