WRT paradoxes... my hypotheses are:
1.
Quantum mechanics. The unobserved universe/timeline is a superposition of all possible universes, none is more true than the others. When you send the bombs back to kill yourself, you have already observed that you are alive, and were not in fact killed by the bombs. This caused the universe to collapse into a state where all the possible timelines, in which the attempt on your life succeeds, are ruled out. I don't know if I explained that very well at all :P.
2.
Another hypothesis (my own, I've never seen it anywhere else yet). There is space-time, which is really like a big 4d block of space that sits there, with our subjective viewpoints moving forward through it. There is also "meta-time" which allows us to move in time. Meta-time is not a dimension, there is no meta-past or meta-future that exists anywhere.
What happens is space-time is like a cellular automaton (but of course not cellular since there aren't any cells :)) which runs according to some rules which we do not know about. It is "meta-time" which allows the automaton to run. As in any other cellular automaton, there are stable states (like four squares in Conway's) which in our universe, actually dominate and spread (cause adjacent cells to become stable). Since they are stable they are logically consistent (no paradoxes), and it is these stable states which give us our apparently unbreakable laws of physics.
When you try to kill yourself in the past, you alter the geometry/topology of space-time (by making a bridge to the past), and to a degree, alter the rules of the machine around the changed geometry. Maybe there are no stable states possible in this geometry and you do cause a paradox, and it does switch between each scenario ad infinitum. However this doesn't happen instantly - the automaton takes time to propogate your effects, sort of like waves in spacetime :P. In another case, there might be a stable state possible, but there is no guarantee it will be anything like the current state of space-time. Maybe the gun jams as the way the automaton resolves itself, or maybe you just erased the Milky Way from the whole of history ;).
Well, this is getting long, so /end.