It's possible Dumbledore knew. During his many conversations with Snape, Snape may have disclosed he had made the Unbreakable Vow with Narcissa Malfoy and J.K. Rowling just didn't show that on page:
Dumbledore listened to Harry’s story with an impassive face. When Harry had finished he did not speak for a few moments, then said, ‘Thank you for telling me this, Harry, but I suggest that you put it out of your mind. I do not think that it is of great importance.’
‘Not of great importance?’ repeated Harry incredulously. ‘Professor, did you understand –?’
‘Yes, Harry, blessed as I am with extraordinary brainpower, I understood everything you told me,’ said Dumbledore, a little sharply. ‘I think you might even consider the possibility that I understood more than you did. Again, I am glad that you have confided in me, but let me reassure you that you have not told me anything that causes me disquiet.’
Half-Blood Prince - pages 335-336 - Bloomsbury - chapter 17, A Sluggish Memory
Canon doesn't tell us if Dumbledore knew Snape had vowed to "carry out the deed the Dark Lord has ordered Draco to perform". Further, it's not implicitly stated that Dumbledore knew that Snape had agreed to put up his own life in order to complete Draco's task to kill Dumbledore.
Deathly Hallows doesn't pinpoint the date that Snape and Dumbledore had the conversation where Snape agreed to Dumbledore's demand that Snape be the one to kill Dumbledore, which was totally separate from the Unbreakable Vow. The conversation takes place on the night Dumbledore mortally wounded himself by putting on the Peverell ring Horcrux, which triggered a fatal curse. Snape was able to trap the curse temporarily in Dumbledore's hand, but they both knew it was a matter of time before Dumbledore would suffer a protracted and painful death. I tend to think this conversation occurred prior to Snape making the Unbreakable Vow, because he suggested Dumbledore go ahead and let Draco kill him -- would he have done this if the Vow had been made?
‘If you don’t mind dying,’ said Snape roughly, ‘why not let Draco do it?’
‘That boy’s soul is not yet so damaged,’ said Dumbledore. ‘I would not have it ripped apart on my account.’
‘And my soul, Dumbledore? Mine?’
‘You alone know whether it will harm your soul to help an old man avoid pain and humiliation,’ said Dumbledore. ‘I ask this one, great favour of you, Severus [...]’
Deathly Hallows - page 548 - Bloomsbury - chapter 33, The Prince's Tale
So at this point, I don't think Dumbledore knew about the Unbreakable Vow because it hasn't been made yet. I do not find Dumbledore to be indifferent to Snape; I find that he is merely suggesting to Snape that only Snape can reconcile in his soul whether killing Dumbledore under the circumstances he is facing would be an act of murder or an act of mercy, and he is allowing Snape the choice to consider it as an act of mercy.
As for the strangeness factor, well, Dumbledore was a strange guy. Who else would give a dinner speech consisting only of "Nitwit! Oddment! Blubber! Tweak!"?