He tells Luke that Vader helped the Emperor hunt down and kill all the Jedi. Well, he sorta did. As far as Obi Wan knows, Anakin did help the Emperor as he set the message in the Temple to tell all the Jedi to return. And Anakin would know where all the Jedi were, too.
How do you know when the construction of the second Death Star began? I've never seen it.
And, the force is strong with the Skywalker's, so Leia could have feasible sensed her mother and her feelings while in the womb.