Here's my thought:
Since temporal causality can work to allow the big-bang to create itself at the start of all existence; can we have a non-universe space outside of the big-bang universe that is drawing the universe to a teleological end?
We discussed this briefly before. I think you've made a number of errors in your thinking.
It is a mistake to suppose that the big bang "create[d] itself." For one, the big bang was an
event, not a thing. A singularity is a feature of a certain mathematical model, not an object in reality with physical dimensions. It wasn't "created." It
happened (according to our reverse extrapolations of the expanding universe).
Second, it requires an uncommon definition of "universe" to suppose that any idea resembling a "non-universe space" makes any kind of sense.
Personally, I think you make too much of the "arrow of time," as if you could establish a preferential or universally inert frame of reference. Time is a dimension just like space (in fact, there is no distinction, there is just spacetime). The coordinates of these dimensions exist simultaneously -- over
there exists simultaneous with over
here, and
then exists simultaneous with
now.
I won't argue that the apparent paradox of the "arrow of time" and the time-symmetry of the laws of physics is not a controversial one, but I simply think you're making unwarranted and dubious inferences. Personally, I think the "arrow of time" appears because of our unique sensory apparatus, and particularly the gaps between our synapses over which sensory data must leap. And again, those are themselves consequences of our relatively uniform velocities within our immediate environment.
Not a vitalism, but some end-state that is outside of time that is impacting reality through transcendence of the temporal arrow of entropy?
If you'd like some better answers to your questions than even I can give you, I can PM you a link to a discussion forum I frequent (less so, lately) where I know a brilliant mathematical physicist that is a regular participant.