No, none of these. Well, something would prevent this from happening, but whatever happened would have a perfectly reasonable explanation at the time.
You have to understand, firstly, that in GR, the universe doesn't start at the beginning of time and slowly evolve until it reaches the end of time. Instead, all of space and time that ever was or ever will be, exist now and forever. And this one big constant eternal space-time is one of many possible solutions to the equations that constrain the universe.
If a potential universe does not satisfy those equations, then it can't exist and therefore it won't exist.
A universe in which your pool ball goes back in time and deflects itself, is not a solution to those equations, and therefore, that will never happen in any GR compliant universe.
So, what might happen instead? Well, you might get bored with the experiment and give up before you get it working. Your girlfriend might come along and tell you that you are neglecting her. Burglars might steal your time machine. Or most likely, you might never invent the time machine to begin with.
It's quite conceivable that in some sense a time machine is theoretically buildable, but the reason our universe is one of the solutions to GR is that no one ever actually builds one in the history of the universe.
Why not, you ask? Because if they did, that universe is much less likely to be a solution to the equations, so we just so happen to find ourselves in one of the more likely universes.
Another way to think about this is that perhaps all possible universes actually exist. In that case, we have to find ourselves in one of the existing universes. We can't find ourselves in a non-existing universe, since non-existing universes don't exist.
However what I don't understand is this -- to say that the bullet shot by A passes B at 4 seconds presumes that we are talking about A's frame of reference (because it is from A's point of view that B is 4 seconds "behind" -- and likewise A is 4 seconds behind from B's perspective). However, what if, from B's frame of reference, A's bullet actually passes B at 0 seconds?
Again, I have not seen that paper in years, so my memory might have just made half of that up.
Sure, tachyons will be strange (different observes won't agree on direction of travel or charge of a tachyonic particle - if we are even able to localize the particle at all; there will be frames of reference where a tachyon won't have energy but will still have momentum, ...), but not really any stranger that what we deal with in quantum mechanics and field theory, and the laws of physics will still hold in all frames of reference.
The community just lost interest in them because our current models of particle physics prevents bare tachyons:
QFT does predict real tachyonic particles. However, they are solutions of unstable vacua, which transit to a tachyon-free phase via symmetry breaking.
Stringtheory predicted tachyons as well, but got rid of them again when superstringtheory came around.
Tachyons are not gone completely, though: Virtual particles can be off-shell and thus tachyonic and feature in the description of self-energy.
Tachyons can also be useful to explain EPR-type quantum 'paradoxa' without having to give up reality.
http://thebestforumever.com/community/threads/tachyon-pistol...
Also, I've never understood how the Novikov self-consistency principle would work in reality. Causality violations can be a lot simpler than killing your grandfather. Say, for example, you set up a pool table with a pool ball and hit it through a time machine such that it went back in time and hit itself, making it miss the time machine? Would the pool ball be suddenly acted on by some invisible force? Would it explode? Would the universe just end? The whole scenario seems so implausible that addressing this theoretical concern is a lot more relevant than worrying about the energy levels required to manipulate space time.