Did Dumbledore know
On the night of 6 June, 1994, Remus Lupin stated that the Ministry "never knew that there used to be three unregistered Animagi running around Hogwarts", the three being his schoolmates Sirius Black, Peter Pettigrew and the late James Potter; Black's, Pettigrew's and Potter's Animagi nature was revealed by Remus Lupin ...
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. Unless we assume Dumbledore was completely insane he did not.
We know that Peter Petigrew told Voldemort about Sirius being an Animagus from the following question: How did Lucius Malfoy know Sirius's Animagus form?
Albus admits in the third book in the last chapter that he had no idea that the Marauders were Animagi. That was part of the reason why Dumbledore had trouble finding the spy in the Order, he had no idea Pettigrew was a rat Animagus.
“A creature I wouldn't want to kiss . . . a spider!” Harry figures out the riddle, and realizes that a spider is about to attack. But what he doesn't realize is what I now believe: that the spider in the maze is Professor Snape, in his Animagus form. It is Spider/Snape who bears down on Cedric and Harry.
Hermione is the brightest witch of her age, and she would be the first in the group to successfully transform into her Animagus form, a rather fitting fox.
He didn't mention that Sirius was an Animagus to Dumbledore because he didn't want to admit to breaking his trust while at school. All in all, Remus did the most to help Harry and the wizarding world.
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.
Strikingly, Harry's Patronus was a stag and his mother Lily's was a doe, a female deer, showing that the family's characters were in harmony and formed part of the same animal group.
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.
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.
Because he couldn't face his guilt and admit to Professor Dumbledore, that he (Remus) betrayed Dumbledore's trust (as the headmaster was the only one-as far as we know- to allow a werewolf in his school, a heaven-sent opportunity) and he didn't conform to Dumbledore's mesures of security (staying put in the Shrieking ...
Sirius's early life proved unhappy; he had come to hate most of his relatives, in particular his mother. He rejected his family's pure-blood elitism and reverence for the Dark Arts. At Hogwarts, rather than be sorted into Slytherin like the rest of his family, Sirius was placed in Gryffindor.
At the end of GOF only a handful of people even know that Sirius is an animagus, let alone his loveable big doggie form. But by the beginning of OOTP we have Lucius Malfoy able to "recognize" Sirius by seeing the large black dog with Harry.
Lily was an animagus too, when Voldemort attacked, she transformed in front of him so that she could attempt to escape. Voldemort hit her with a spell which locked her in her form as a twisted way of leaving her alive for Snape. Harry never noticed but there was always an animal watching him as he grew up.
Bellatrix Lestrange Killed her own cousin, Sirius Black: She killed Sirius (who was her cousin) during the battle at the Department of Mysteries. She was remorseless and was cackling in glee after she had murdered Sirius.
Amelia Black, the daughter of alleged mass murderer Sirius Black and Sierra Snape, Severus Snape's sister, struggles most of her life to get her father's attention.
Harry had been working on his Patronus charm, though, and Malfoy & Co. got what was coming to them. One of the most notable Malfoy moments is where he hints that he knows all about Sirius Black (7.1. 48).
Harry discovers this in Prisoner of Azkaban, when his Patronus turned out to be a stag, his father's Animagus form.
Sirius Black had a lot of love in his life. Mainly for his best friends but still. One of the greatest loves came not from his connection to James Potter but rather with his relationship to Remus Lupin.
Because there was no need for him to become one. Besides he had lot of other important stuff going on in school unlike his father and his friends who were determined to help Lupin.
For example: if Viktor Krum can transfigure his head into a shark's, like he does in Goblet of Fire, that doesn't mean he is an Animagus. Transfiguration is a form of spell-casting, whereas becoming an Animagus is forever. And yes, he could turn into a falcon.
Ron (Ant that has wings) - Loyal to the queen, and willing to die to save the queen as well. Also quite social.
Nymphadora Tonks, who we first meet in Harry Potter and the Order of the Phoenix, is a Metamorphmagus. She tells Harry that the ability helped her greatly in the Concealment and Disguise in her Auror training.