I guess it is possible that the universe could have undergone a Big Crunch prior to the Big Bang. Itís really quite hard to tell these things because information about the state of the universe before the Big Bang is erased in the process. One of the possible hypotheses advertised a few years back is that the universe continually explodes and then recollapses under its own weight, only to explode again. And again.
The most recent observations of how fast the universe is expanding and how clumpy the stuff in it is seem to indicate that we are not in such a cycle. The expansion of the universe, if anything, appears to be accelerating! Of course people could find out later that the measurements were wrong, but more and more independent ways of looking at things seem to be indicating that the universe will keep expanding, and even do it faster.
This still doesnít tell us about previous Big Crunches, it only indicates that we donít believe one is coming up.
As for your second question, Iím not too sure what it means to "live forwards" and what it means for a world to "travel backwards" in time. Time is a number you need, in addition to three numbers for space, to identify where and when something happens.
One of the principles of physics that doesnít seem to have any experimental evidence against is that of causality. An event can only affect events in the future, and can have no effect on prior events. This principle seems to apply to all interactions of all kinds of stuff with no exceptions, so "we" and "the world" should agree on which way in time the future is.
(published on 10/22/2007)