Did Darth Vader Know He Had Twins?

At the end of ROTJ Vader senses out of Luke that he's worried about his 'sister', and deduces he had twins with Padme. They fight, Vader kills the Emperor, which in turn nearly kills Vader. At the end, after his mask is pulled off Vader says "Tell your sister, you were right."

Does Darth Vader ever find out Leia is his daughter?

Vader discovers Leia is his daughter because of Luke.

It's not until Return of the Jedi, when Luke confronts Vader and the Emperor in the third act of the film, that the Sith Lord realizes his son is there to protect his twin sister.

Did Anakin know he was having twins?

Leia is raised by Senator Bail Organa and Queen Breha Organa on Alderaan. Anakin, who has since become the Sith Lord Darth Vader, is unaware of his children's birth. The twins soon became part of the Rebel Alliance following the deaths of their adoptive families at the hands of the Empire.