Question 1: What is so special about "Move into the fast lane"?
"Move into the fast lane" is far more popular than the other sentences because it has a common figurative meaning, rather than just a literal meaning about traffic. Specifically, if someone is "in the fast lane" or "living in the fast lane" we mean that they have an especially exciting and fast-paced, perhaps even dangerous, life. From Cambridge Dictionary:
life in the fast lane
a way of living that is full of excitement, activity, and often danger:
Parties and women - he was living life in the fast lane.
We can also use "fast lane" even more abstractly, for anything that is in a fast and exciting mode. So while some examples of "move into the fast lane" are talking about actual automobile traffic, more are probably speaking figuratively about someone making a change to a more exciting life. Running a search for "move into the fast lane" in Google's Incognito Mode (to minimize skewing the search results), the top hits I see are about
"Life in the slow lane" also occurs occasionally, as a contrast to "life in the fast lane", but it's less common as an idiom.
Question 2: Is there any difference between "pull into" and "move into"?
There is a slight connotation of "slowly" and maybe "carefully" with "pull into" that is absent from "move into". In fact, in the abstract, Cambridge Dictionary defines the phrasal verb as
pull in/pull into somewhere
— phrasal verb with pull UK /pʊl/ US /pʊl/ verb
If a vehicle pulls in or pulls into somewhere, it moves in that direction and stops there:
He pulled in at the side of the road.
I pulled into the empty parking space.
Obviously, when one "pulls into the fast lane" or even "pulls into the slow lane" you aren't bringing the car to a stop; however, there is still a slight suggestion that the maneuver is fairly gradual and also probably smooth, which the plain "move into" doesn't convey.
Question 3: What are the right/common sentences to use in these cases?
Any of the four sentences are fine, depending on what you want to convey. "Move into the fast lane" is much more common than "pull into the fast lane" for the figurative usage, because that kind of shift is probably not particularly gradual. I suspect it's also more common for the literal meaning of the term, for the same reason. That pattern is slightly reversed for "into the slow lane", because there one is going slower.