Did kos-mos die on xenosaga 3

There's no way we can be sure about this because they canceled any plans for the franchise... meaning no Xenosaga IV. If I really had to guess though, I would say that she is not dead and cannot fully die as long as the main part of her system (head and torso) survives. Chances are, she can be repaired, though with much difficulty considering how damaged she was at the end of the game.