#5, the empire strikes back.
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Vader's alter-ego is Anakin Skywalker. Then the Emperor told Vader that Luke is the off-spring of Anakin SkywalkerWell, in The Empire Strikes Back, in the battle of hoth, when Vader arrives at the ice planet, he says "The rebels are here. And I believe Skywalker is with them." He doesn't make the connection that he is Anakin Skywalker. He does think, however, that the one who took down the Death Star in A New Hope was special and he felt the force in him. After the battle of hoth, Vader spoke with the Emperor, and he tells Vader that Luke is the son of Anakin Skywalker. The Emperor said that he must be destroyed. I believe Vader was trying to save his son in saying "He could be a powerful ally." Then he offered Luke a rule as father and son, and Luke refused. Vader didn't find that Leia was his daughter until he felt the connection with her and Luke in the Emperor's throne room in ROTJ.
Yes, they discover that in episode VI, Return of the Jedi
Yes. You find out in Return of the Jedi (VI).
It could be argued both ways. But directly he did not. He convinced his father to come back to the good side. When he did Darth Vader killed the emperer. That caused Darth Vader to die because it was only through the emperer's power that he was being kept alive. He was injuring by lava in his fight against obi whan. So when he killed the emperer Darth Vader died. So technically nobody killed Darth Vader he just died. However it also could be argued Luke is to blame. If he had not convinced Darth Vader to come back to the light side of the force he would not have died.
Since Return of the Jedi takes place 4 years after A New Hope and A New Hope takes place 19 years after Revenge of the Sith, Luke Skywalker is 23 years old.