Darth Vader never knew he had twins until that final battle on the second Death Star. The force ghost of Obi Wan had warned Luke to bury his understanding of Leia deep or Vader would be able to sense his feelings.
However, Leia's strong-mindedness during Vader's probe denied him the opportunity to discover she was his daughter. That serves to justify Vader's lack of awareness of his connection to Leia, although, perhaps, not as satisfyingly as if Lucas had planned for the relationship from the beginning.
However, she chose not to tell Anakin, presumably because he was about to go into a battle. Anakin was prone to letting his emotions affect him, and knowing that Padme was pregnant would've thrown him completely off the task. And Padme probably didn't think that her husband would be fighting battles for months to come.
In-universe, we can say that Vader sensed no deception on the part of The Emperor, and therefore assumed he was telling the truth about Padme's death. The death of the child (Vader knew she had been pregnant, just not with twins), on the other hand, was never mentioned.
In The Phantom Menace, when Qui-Gon Jinn meets tiny Anakin Skywalker and his mother, he goes on about how the Force is like, crazy strong with Anakin or whatever, and then asks his mother about Anakin's dad. Shmi Skywalker is pretty clear in her answer: There isn't a father. She was impregnated by the Force.
After all, he only learns of Padme's pregnancy in a brief moment after rescuing the most powerful man in the known galaxy.
Darth Vader only learned he had a daughter in Return of the Jedi - but there's no evidence he realized she was Leia before his death and redemption. Darth Vader may have never known Princess Leia was his daughter in Star Wars and never cultivated a relationship with her, even when he was a Force ghost.
Padmé died shortly after giving birth to Luke and Leia, swearing with her last words that there was still good in Anakin Skywalker. The twins were split up, with Luke going to Tatooine with Obi-Wan while Bail took Leia to Alderaan.
To hide the birth of her twins from both Darth Vader and more importantly Palpatine. The fact that Amidala had given birth to twins was a secret known only to a few after her death. The mortician Commodex Tahn used a hologram to make her still appear pregnant during her funeral.
Nope. Luke, unlike Leia, never really had a father figure in the first place. Leia never was saved by Darth Vader the way Luke was, and never understood Luke's forgiving him. She hid her identity as Darth Vader's daughter, and identified as Bail Organa's daughter.
Before the Disney acquisition of Star Wars, Vader only found out about Luke after Palpatine told him that he was the son of Anakin Skywalker. But the post-Disney acquisition era changed all of that, especially in the comics, where Vader discovered before Palpatine ever did that Luke was actually his son.
Reva knows that Anakin Skywalker is Darth Vader because she was one of his targets—a child—during the Order 66 killings that took place during the events of Star Wars Episode III: Revenge of the Sith.
Even though he knew of Anakin's relationship with Padme and the fact that she was pregnant, there is nothing to suggest that Palpatine was aware that Padme was pregnant with twins, as opposed to one child, in the first place.
At some point after his transformation into the armored Darth Vader, Amidala's husband, the fallen Jedi Knight Anakin Skywalker came to visit the mausoleum, stricken by grief and remorse for his part in her death.
The short answer is yes, without a doubt, Vader regretted becoming a cybernetic monstrosity. First, there's the constant pain and discomfort from his suit. Worse, however, is the psychological pain. Vader destroyed everything he had ever known for a chance to save Padme.
Of course, Obi-Wan and Yoda couldn't attend the funeral, knowing that Palpatine was likely keeping a close eye on the proceedings. After all, keeping Anakin as Darth Vader relied on him believing he killed Padme, so the Emperor needed to make sure no part of the funeral suggested otherwise.
Arguably, Vader loved Padmé even more than Anakin did. After what he did to Padmé on Mustafar, Vader was obviously devastated. The worst part was, he believed it was his own fault, even though the choking only had a very indirect influence on Padmé's death.
Some people believed it was Breha who Leia was remembering on Endor, thinking she was her biological mother. However, this didn't end up being true as the prequels established Padmé as her (and Luke's) mother.
Anakin saw his mother again in a Force vision around 21 BBY on Mortis. The vision was induced by the Son, one of the powerful Force-wielders controlling the planet, through which the current of the Force flowed.
It's not shown, but it is clearly insinuated that Anakin killed the younglings. This is confirmed later in the movie when Obi-Wan informs Padme that Anakin killed them. Both are torn apart with hurt and confusion, trying to make sense of something so terrible. How could Anakin do such a thing?
The marriage, witnessed by the droids C-3PO and R2-D2, was kept secret due to Amidala's place as a prominent Senator in the Galactic Republic and because of Skywalker's membership in the Jedi Order; the Jedi Code prevented Jedi from falling in love and forming attachments such as marriage.