Six was stabbed by a Sangheili Zealot when the Elite was drawing his Energy Dagger. Actually It wasn't just a Sangheili Zealot that killed him, the last elite that stepped into the screen before he died, was the Arbiter.

Did the arbiter kill Kat?

nope. Killed by the same Field Marshall that accompanies the Zealots in Winter Contingency and takes the person hostage, and that kills Kat.

Could Noble 6 have survived?

How would you expect to survive that? It's nearly impossible. Keyes' last words to Noble Six (face-to-face) are "Good luck to you too, Spartan", indicating that he knows that Six, after his decision to sacrifice himself, would die.