Let's make this right.
T 800 Is not the original one. He's just a replacement.
Regarding to Connor's death, lemme tell you this:
A T 800 Kills Connor in the future (2032, if I remember well). That's the same that takes part in T3, not in T1 nor T2. He was captured by the resistance after he killed Connor, reprogrammed by Brewster (Connor's wife and the 2nd in charge) and sent back to the past, to protect Brewster, and his primary objective, Connor.
If there's something you should've learnt about the T Saga, is that the future is meant to happen one way or another. There's no way to avoid that. The T 800 was selected by skynet to kill Connor because that MODEL had emotional reactions in Connor (Remember Connor said "He's the closest I ever had to a father".
Ressuming, just at about the end the T800 said "We we'll see again" (or something like that), that should make you realise that he's going to meet with Connor in the future (probably, The T 800 was mentioning the moment before he killed Connor), so the T 800 IS going to kill Connor.
Period.
