Ask a Foolish Question by Robert Sheckley appears to tick most of your boxes. It's also an (unaccepted) answer to this question. It can be found and read it at Project Gutenberg (thanks to user14111 for the link), and on the Internet Archive. It has humans and aliens searching for an entity that knows the answer to every question but every question asked of him is not valid.
Originally published in Science Fiction Stories, #1 1953, the story opens describing the Answerer, who is ancient and was created by a race who "Knew".
Answerer was built to last as long as was necessary — which was quite long, as some races judge time, and not long at all, according to others. But to Answerer, it was just long enough.
Of the race that built him, the less said the better. They also Knew, and never said whether they found the knowledge pleasant.
They built Answerer as a service to less-sophisticated races, and departed in a unique manner. Where they went only Answerer knows. Because Answerer knows everything.
It contains humans searching for the Answerer, and one of the men searching is wanting to ask about life and death:
“We’ll find out,” Morran murmured. He helped the old man unstrap himself; “We’re going to find the Answerer!
“Yes,” Lingman said. He pulled himself to the vision plate and looked out on the bleak prairie of the illusory sub-space. He was a biologist and an old man. He had two questions.
What is life?
What is death?
There is also a group aliens who are are "hunting purple". Apparently they have no idea why they are collecting purple:
The great job of Lek and his kind was the gathering of purple. They found purple imbedded in many parts of the fabric of space, minute quantities of it. Slowly, they were building a huge mound of it. What the mound was for, no one knew.
The alien Lek says he's going to ask the Answerer why they are gathering purple:
“I suppose you’ll ask him what purple is?" Ilm asked, pushing a star out of his way and lying down.
“I will,” Lek said. “We have continued in ignorance too long. We must know the true nature of purple, and its meaning in the scheme of things. We must know why it governs our lives.” For this speech Lek switched to Ilgret, the language of incipient-knowledge.
Lek does get to ask his question to the Answerer, but the Answerer cannot answer because Lek failed to ask the real question. The following passage also indicates that the Answerer does know the answer, but cannot answer due to "a greater explanation". It also hints at the possibility he's sentient, as he mulls over the answers he's not allowed to give:
“Come now,” Lek muttered, his pride hurt. “You can do better than that. Now then. The purpose of my kind is to gather purple, and to build a mound of it. Can you tell me the real meaning of this?”
“Your question is without meaning,” Answerer said. He knew what purple actually was, and what the mound was for. But the explanation was concealed in a greater explanation. Without this, Lek’s question was inexplicable, and Lek had failed to ask the real question.
The proper questions. The race which built Answerer should have taken that into account, Answerer thought They should have made some allowance for semantic nonsense, allowed him to attempt an unravelling.
Answerer contented himself with muttering the answers to himself.
At the end of the story the requirements to asking the "right question" are revealed. In order to ask a question you must
already know most of the answer.