Interesting topic.
I am in the very sceptical camp, when it comes to (backward*) time travel, which I think would entail far too many paradoxes for a reality to fulfill the coherence it needs to work. I have yet to see a time travel story where I have not needed to suspend disbelief at all, which tells me a great deal about the (paradoxical) nature of time travel.
Most of these problem have been pointed out already. One point bothers me in particular, though, and it is ignored in movie narratives such as Terminator and The Looper what makes the "first"** interference from the future to any one point in time final. If we look at Terminator, and one terminator (T1) is sent back to time X from time Y to kill John Connor (JC), and it succeeds to destroy the resistance's terminator (T2) and JC, why can't the resistance send back another terminator (T3), to time X, or even time Y, to destroy T1 before it succeeds
Ok, one might argue (rightly or wrongly) that the future resistance would not know about JC's importance (or likely, would not exist) if JC is taken out at an earlier point, but the point stand for narratives following this structure. If two future parties have opposing interests in the outcome of one particular event, they could just send more and more agents/forces back in time to point X. What would close point X in time off from further future interference, so there could be a final outcome?
Confusing? Probably, yes, quite a bit. However, this confusing state of matters does very much seem to fit into the convoluted mess created by backward time travel.
All in all, I agree with Fishbowl Helmet on time travel, save for the part about big bang, which doesn't seem to fit into what I have learned about the latter subject. It is more than just a big explosion, whose radius travellers could be outside. Space itself is expanding since big bang, rather than galaxies moving outwards in space from some center point. If we travelled back in time to the very start (and, for that matter, that were even possible), we would find ourselves within that singularity (scale factor = 0, and space itself does not yet exist), which would certainly not be able to contain living beings in their current form... Let us just say it would be a bit less attractive than curiosity would suggest.
Other than that, good posts, Fishbowl.
As for the specific rule of time travel in the OP, that it is "time train" rather than "time automobile" (or something even more flexible, like "time airplane"), it is an a decent idea, although this restriction was not made clear from the start. I agree that a time traveller would indeed probably come out of such a machine as it is activated for the first time in history.
While a completely arbitrary restriction, the writer may indeed make such a rule. However, it doesn't really solve any paradoxes.
By the way, I like Dr. Who's way of making paradoxes created by time travel harmful to the structural integrity of the universe. That very much seems to be the way it should be, in my mind.
Like J Riff, I am less impressed by Looper, writing-wise. Sure, the actors are quite good, but the story leaves a lot to be desired, and in particular the way it deals with this subject
By the way, sorry if I sound like just another negative voice. I do not mean to discourage, but this is a difficult plot point to get right.
* Time travel only forward obviously creates no paradoxes, as it doesn't create duplications of any point in time.
** First and last may be a bit hard to define in a universe where backward time travel is possible, as time is no longer a straight line. I refer to the first in sequence, here. The second time someone travels back to point X would be in from the "alternative universe" created by the first backwards time travel (to point X) but may actually be a shorter backwards leap in linear time (as in first being from 2465 to 2432 and second being from 2447 to 2432 in the changed timeline).