There are, as I understand it, three possible ways to look at time-travel.

You can believe in a "fixed" timeline, i.e. no events can be changed by going back in the past - so any event undertaken in the past after time-travel is already part of the "future" reality in which you live. An example of this is Harry Potter and the Prisoner of Azkaban - when they go back in time, everything they do had occurred when they originally went through the events (like Harry being the one to cast the patronus from the far side of the lake.) In this case, the very fact that you're alive to be able to time-travel backwards demonstrates that you either don't try, or try and fail to, kill yourself.

You can believe in a "dynamic" timeline, in which events are not fixed, and you can change them by travelling back in the past. In this case, the paradoxes you guys are discussing arise.

Or, thirdly, you can believe in a sort of multiverse theory, in which either new universes or new timelines are spawned each time you do something - and so when you go back and kill yourself, you're only affecting certain timelines, whereas in others you continue to survive. A certain requirement of this is that you can't travel between timelines/universes - otherwise the paradoxes we've already observed can come into effect.