Two questions: First, I’m sure he did enough of a background check to know that Uncle Ben is dead, and killed by a burglar. But does Stark know Peter could have stopped the burglar beforehand? Peter sort of hints at it, but never explains it directly. “When you can do what I do, and the bad stuff happens, and you didn’t do anything, it happens because of you.”
From what we see in Homecoming, Tony doesn’t spend a lot of time with Peter while in Germany. And when he’s yelling at Peter for the near-fatal ferry disaster, he never brings up Ben’s death either.
My guess: Tony doesn’t know about Peter’s contact with the burglar.
Second question: Does Tony know how Peter got his powers? In Civil War, Peter starts to tell him, but Tony cuts him off. Here again, we don’t know how much they spoke on the plane, but it was probably mostly about Captain America. And Tony’s a scientist who’d be fascinated by what that spider bite did, but he never brings it up. (And not just Stark–look how many people have been desperate to get the Super Soldier formula.)
My guess: Tony thinks Peter’s an Inhuman, who’s also an inventive genius who invented the webbing formula. I wonder if this might come up in Infinity War.
Peter has no problem telling Ned about the spider bite with practically no prompting. I doubt Tony Stark could stop nerdy motormouth Peter Parker from blabbing about it to him (his sort of father figure) in every detail even if he wanted to.
I’d guess that Stark knows about the spider bite and Peter told him off screen. Just part of the “We’re not rehashing this origin story yet again” thing. Stark made several suits for Parker and you’d want to know the circumstances of someone’s powers so you could tailor the suit to complement them.
No idea if Stark knows about Ben’ murder but all them heroes have tragic tales anyway so I doubt it would change anything.