did Dumbledore know that Draco Malfoy was assigned by Voldemort to kill Albus Dumbledore?
Yes. As we see in Snape's memories at the end of Deathly Hallows Dumbledore spoke about it with Snape before the school year even started:
"I refer to the plan Lord Voldemort is revolving around me. His plan
to have the poor Malfoy boy murder me."
Did Dumbledore know that Draco Malfoy placed the poisoned mead in the wine, and did he know that Draco Malfoy placed Madam Rosmerta under imperius curse, which placed imperius curse on Katie Bell, in attempt to deliver the cursed necklace to Dumbledore?
He knew that Malfoy was behind the poison and the necklace, as he mentions this to Malfoy right before he dies:
“Oh yes, I do,” said Dumbledore mildly. “You almost killed Katie Bell
and Ronald Weasley. You have been trying, with increasing
desperation, to kill me all year. Forgive me, Draco, but they have
been feeble attempts. ... So feeble, to be honest, that I wonder
whether your heart has been really in it.”
As to how he knew this, there are several possibilities.
- Malfoy was the obvious suspect.
- Snape who was stalking Malfoy on Dumbledore's orders might have told him.
- He might have believed Harry's accusations.
- Dumbledore in general knows most of what goes on.
He apparently did not know, though, that Madam Rosmerta was under the Imperius Curse (or else he was a really good actor), as he only figures this out in the conversation with Malfoy right before he dies:
But you must have had an accomplice, all the same ... someone in
Hogsmeade, someone who was able to slip Katie the — the — aaaah
Dumbledore closed his eyes again and nodded, as though he was about to
fall asleep. “... of course ... Rosmerta. How long has she been under
the Imperius Curse?”
“Got there at last, have you?” Malfoy taunted.
If Dumbledore knew all this, why didn't he ban Draco Malfoy, Vincent Crabbe and Gregory Goyle from returning to Hogwarts? Why didn't Dumbledore ask the Ministry to arrest Draco Malfoy at the first place?
Dumbledore was trying to protect Malfoy. From a conversation with Snape (in Snape's memories):
Scowling, Snape said, “The Dark Lord does not expect Draco to
succeed. This is merely punishment for Lucius’s recent failures. Slow
torture for Draco’s parents, while they watch him fail and pay the
price.”
“In short, the boy has had a death sentence pronounced upon him as
surely as I have,” said Dumbledore.
From his last conversation with Malfoy:
“I haven’t got any options!” said Malfoy, and he was suddenly white as
Dumbledore. “I’ve got to do it! He’ll kill me! He’ll kill my whole
family!”
“I appreciate the difficulty of your position,” said Dumbledore. “Why
else do you think I have not confronted you before now? Because I
knew that you would have been murdered if Lord Voldemort realized
that I suspected you.”
Essentially Dumbledore had to pretend that everything was fine, for Malfoy's sake. However, so as not to put other people's lives at risk he enlisted Snape to keep an eye on Malfoy (again, from Snape's memories):
“Good. Now then. Your first priority will be to discover what Draco is
up to. A frightened teenage boy is a danger to others as well as to
himself. Offer him help and guidance, he ought to accept, he likes
you —”
“— much less since his father has lost favor. Draco blames me, he
thinks I have usurped Lucius’s position.”
“All the same, try. I am concerned less for myself than for accidental
victims of whatever schemes might occur to the boy. Ultimately, of
course, there is only one thing to be done if we are to save him from
Lord Voldemort’s wrath.”
So in short, from Dumbledore's perspective actively preventing Malfoy from completing his mission was not a viable option, while at the same time letting him continue was not such a big risk. Indeed, in the end nobody (besides Dumbledore) came to much harm because of this, as Dumbledore himself told Malfoy:
No harm has been done, you have hurt nobody, though you are very
lucky that your unintentional victims survived. ... I can help you,
Draco.”
