Did Severus make the unbreakable vow before 또는 after Dumbledore asked him to kill him?

Good question. I had wondered myself that often.

I think Severus did the Vow after he knew about Albus' condition. He had no qualms in making the Vow with Narcissa. His hesitations were only apparent. This is a quote from HBP, Chapter 2, when Severus replied to Bellatrix who suspected him to be a spy against the Dark Lord :

"I am pleased to say, however, that Dumbledore is growing old. The duel with the Dark Lord last
월 shook him. He has since sustained a serious injury because his reactions are slower than
they once were."

The serious injury refers to the curse Dumbledore got when he put on the Gaunt ring. Therefore, it replies to your question. In DH, in the Prince's Tale chapter, it is described that Severus had come quickly to Dumbledore to treat the curse. He said that he could contain it for about a year. Then there's a whole conversation between Dumbledore and Severus, when he asked him to kill him to spare Draco's soul. At this point, not only they had both arranged the mercy killing of Dumbledore, but they knew about Voldemort's plans of sending young Malfoy to kill the old Headmaster.

Read the books, all the 답변 are there. Even if I must concede that I did need several readings to reply 질문 such as the one 당신 asked. Some things need time to get assimilated.