Why didn't Kellog die in S1E5?

The purpose of the experiment was to prove or disprove the grandfather paradox theory of time travel.

In this case, the result was inconclusive, because it's never stated for certain that she was the right person and not just an innocent victim who shared the name. Remember they were going after multiple people with the same name in the hopes of finding the right one.

On the other hand, if they'd been successful, Kellog still wouldn't have "died" -- he simply would never have existed in the first place. Which creates the paradox: if Kellog never existed, then Liber8 wouldn't have needed to kill anyone to make him not exist, and so he would have existed, and so on.

The ultimate point here is that none of the characters know the rules/logic/physics involved, and this was one way they were trying to find out.

It's equally possible that through that death, they've already skewed off a parallel timeline completely separate from the one they left, and they are now incapable of returning to that one because it no longer exists. (The Back to the Future theory of timestreams.)

UPDATE: Given what we've seen in Season 3 so far, there are several distinct possibilities (Spoilers follow):

• The easiest explanation is that she was the wrong victim. In this case, there is no paradox at all, and the universe keeps on ticking. We've already seen how both Kiera and Liber8 have made themselves part of this timeline, so there is likely a form of the self-consistency principle at work. So, it's likely that in the "original" timeline, the woman was killed, and her murderer never caught. Only by backtracking do we realize that the killer was actually from Liber8, so the timeline remains consistent and paradox is not only avoided, but impossible.

• The woman was the right woman, but temporal inertia hasn't yet caught up with the timeline.

When Alec went back to save Emily, the timeline didn't start to crumble until some time after he made the change; long enough for Kiera to be questioned, escape, be re-captured and then recruited. So given that time could continue on for a period following the divergence, it's possible that the time differential between the woman's death and when it would have had an impact on the timeline was longer, and so the timeline hasn't yet diverged. Though how/if this would matter given the timeline divergence Alec caused is unclear.

• The woman was the right woman, but the change wasn't drastic enough to shatter the timeline.

Obviously, only severe timeline changes are disastrous; otherwise, every conversation Kiera had with anyone would have caused fracturing since in the "original" timeline, they couldn't have taken place. Plus, we know the "freelancers" have been coming back for some time in order to put things in place for their missions. Time is clearly able to absorb a certain amount of change, like having the uncaught serial killer be caught, so there is definitely some elasticity. Since Alec Sadler is a major player with a serious amount of influence in the future, a change to his past has more devastating effect than a minor player like Kellog, so one completely shatters the timeline while the other is absorbed and the universe looks the other way. As to how he would remain alive: he's just an orphaned piece of a defunct timeline, just like the "duplicate" Alec and Keira are.

• Only direct influence in one's own past is strong enough to cause divergence.

Kellog wasn't directly responsible for his grandmother's death, and in fact had no foreknowledge of it having happened, but Alec's tampering was specifically to alter his own past. In this scenario, the paradox isn't the change itself but the fact that one is altering events inside one's own memory. If you go back in time to prevent Event A, then it never happens, and so why do you need to travel back to stop it? This is different from the situation with Kellog's grandmother because no one involved had witnessed the original events. Now, suppose Kellog were to get hold of the device and travel back in time to save her. That would cause a timeline-shattering paradox because he is directly altering events that he has witnessed and that are part of his personal history.

Unfortunately, we still don't have enough knowledge of the time travel mechanics in this world (e.g., Keira's "lifeboat") to know for certain what's going on. I suspect the writers are making up a lot of it as they go along rather than having a full vision already mapped out. For the moment, we don't really have anything more certain than "wibbly wobbly timey wimey".

Everything these travelers do affects the timeline. Everyone they kill affects not just the loss of their offspring but anyone they influenced. There is no paradox. It just means time corcled back and took off in another direction. The old timeline never disappeared. In theory that timeline and the original future is part of the past. Time never breaks it just circles back and creates a new world much like stiching up a seam. Now if the writers decide to create a paradox then the program is lost and just becomes entertainment with no scientific thought. Which most shows on time travel become.