Snape pretends to not understand Harry and carries on like nothing happened. We know, however, that he would understand what Padfoot meant and, being the clever guy he is, he could deduce that Harry believed Sirius was being held captive at the ministry.
Fortunately, he explains that he has no more to give her and turns to go. As he does, Harry sends tells him that "he has Padfoot at the place where it's hidden," which is meant to warn him about Sirius and the Ministry of Magic.
He would always resent Harry for being James' son instead of his. Snape even said that he didn't want anyone to find out that he was protecting Lily's child, “especially Potter's son” (pg. 679 of Harry Potter and the Deathly Hallows) because of his hatred for James.
No one knows the truth about Sirius Black yet, not even Severus Snape. At that time, he didn't know that Sirius Black was innocent. At that time, he knew what everyone else thought they knew - that Sirius was a dangerous criminal that was out to get Harry Potter and that was said to be Voldemort's right-hand.
The Death Eaters were divided into cells, so each only knew the identities of a few of the others - and it's pretty clear JK Rowling wrote them that way *in order* to explain why Snape didn't know either that Peter was a Death Eater or that Sirius wasn't, as this detail doesn't seem to serve any other plot purpose.
The answer is that Sirius wanted to deflect attention from the real Secret-Keeper. He probably hoped Voldemort would go after him so he could steer attention away from Pettigrew.
Image via Warner Bros. Similarly to Harry, Snape's patronus represents a person. In the memories he gives Harry before his death, Snape shows Dumbledore his patronus, using the form to prove his love of Lily Potter, Harry's mother.
While in school together, Sirius once tricked Snape into almost entering the Shrieking Shack while Lupin was there, transformed into a werewolf. James realised the danger and stopped Snape, saving his life; this is the incident Dumbledore referred to at the end of the first book.
In GoF, Sirius doesn't believe that Snape was a death eater. He says that he couldn't see Dumbledore allowing him to work at Hogwarts if he'd ever worked for Voldemort. But after Voldemort's resurrection, most people seem to know of Snape's status as a spy.
Snape definitely knew that Sirius was an animagus after PoA since he was in the room while Lupin was telling the story about him becoming a werewolf and how his friends became Animagus. Snape also has the added incentive that he hates Sirius' guts. So they would both have handed in Sirius if they'd known where he was.
The acclaimed author explained that Harry paid tribute to Snape because of “forgiveness and gratitude.” Though Snape bullied Harry, he also saved him. “Harry hoped in his heart that he too would be forgiven.
"The first thing Snape asks Harry is "Potter! What would I get if I added powdered root of asphodel to an infusion of wormwood?" According to Victorian Flower Language, asphodel is a type of lily meaning 'My regrets follow you to the grave' and wormwood means 'absence' and also typically symbolised bitter sorrow.
Voldemort thought Snape was the owner of the Elder Wand. Using Avada Kedavra while the Elder Wand was resisting Voldemort was already a risky business, but using it against the true owner would be suicide (as Voldemort would find out when he used the Elder Wand against its true owner: Harry).
Harry inherits all of Sirius's possessions, including the house at Grimmauld Place, the house-elf Kreacher, and Buckbeak the Hippogriff.
He said, "According to Victorian flower language, asphodel is a type of lily meaning 'my regrets follow you to the grave' and wormwood means 'absence' and also typically symbolizes bitter sorrow." If you combined that, Snape's words mean "I bitterly regret Lily's death."
It is hinted that he has done something that would inspire Harry to seek revenge, but it is not until Christmas that Harry learns that Sirius betrayed Harry's parents to Voldemort, then, when confronted by their friend Peter Pettigrew, killed Pettigrew and twelve incidental Muggles.
Voldemort intentionally made six Horcruxes, but when he used Avada Kedavra on Harry, he unintentionally created a seventh Horcrux. Instead of dying, Lily's love for Harry created a counter 'curse' known as Sacrificial Protection and saved Harry.
Dumbledore didn't know for certain that Sirius was innocent, and for the most part, it didn't matter. Sirius wasn't useful. Also, see Remus - Dumbledore saw no use for him, so didn't help him.
So Dumbledore didn't want to risk Draco and Snape to be killed and also needed Snape to continue working for Voldemort. Snape working for Voldemort while also helping Harry and the DA was an essential piece of the plan.
The short and simple answer to your question is “no”. Dumbledore - like the rest of the wizarding world - did not know that Peter Pettigrew set up Sirius to take the fall for his crimes.
He did not wish Sirius' death. He did what he could to protect him. In spite of what Harry chose to believe, Sirius' decision wasn't Severus' fault. Sorry to lose a Member of the Order he would have been - they weren't that many; but it was not a personal loss.
Snape hears Lupin admit this while hiding under the Invisibility Cloak and also hears Lupin explain that he was not part of Sirius's joke. Snape still thinks Lupin is helping Sirius, however, and attacks Lupin. He then lets slip that Lupin is a werewolf the next morning.
Yes, Dumbledore did ask Snape to kill him so Draco wouldn't. Dumbledore did indeed ask Snape to kill him, so that Draco wouldn't either succeed and maim his own soul, or more likely fail and be killed by the Dark Lord for his almost inevitable failure. "All the same, try.
1 Albatross
As the rarest Patronus in the Wizarding World possible, the Albatross represents an enormous bird that flies freely across any sea. This rare Patronus represents those who are simultaneously fearless, optimistic, happy-go-lucky, and ambitious.
When Snape used his Patronus to lead Harry to the Sword of Gryffindor. But of course, it was not just Harry whose Patronus saved the day.