From the Kaku site I posted above ...

"Interestingly enough, Stephen Hawking once opposed the idea of time travel. He even claimed he had “empirical” evidence against it. If time travel existed, he said, then we would have been visited by tourists from the future. Since we see no tourists from the future, ergo: time travel is not possible. Because of the enormous amount of work done by theoretical physicists within the last 5 years or so, Hawking has since changed his mind, and now believes that time travel is possible (although not necessarily practical). (Furthermore, perhaps we are simply not very interesting to these tourists from the future. Anyone who can harness the power of a star would consider us to be very primitive. Imagine your friends coming across an ant hill. Would they bend down to the ants and give them trinkets, books, medicine, and power? Or would some of your friends have the strange urge to step on a few of them?)

In conclusion, don’t turn someone away who knocks at your door one day and claims to be your future great-great-great grandchild. They may be right."

You just made me think of the last x-men franchise...
If time travel did indeed become physically feasible... It does not have to be practical, for it to be exploited...
Would time travel be a matter of pure physics and not psyche?

We don't know. The math says time works in either direction.

If some form of monistic idealism accurately maps the state of reality then psyche could work (by psyche I assume you mean mind). If the universe is compatible with a materialist framework however time travel will be a feat achievable only through the application of advanced theories in physics. The metaphysics of reality is not knowable with absolute certainty by any current human standards. That being said planes fly, bombs explode, satellites communicate the wonders of man and the universe every second of the day, therefore, it seems like the materialist postulation of metaphysics is the most reliable.

As far as I can tell time is analogous with change. This answer is very basic but if time is not a fundamental dimension of space time then why is the theory of relativity so successful? If we remove time as the 4th dimension the theory of special relativity can no longer make sense of the time dilation we routinely observe around black holes and neutron stars. If you can not come up with a better theory than Einstein for why this occurs then you must accept that the reality of time is at very least the most plausible physical understanding of the universe humans are currently capable of employing.

The Hawking Theory you guys are referring to is called the Hawking Chronology Protection Conjecture. Also a reason we might not be inundated with time travelers right now is that many theoretical time machines allow for time travelers to only travel as far back as when the time machine was first turned on or created (and we haven't created a time machine yet).

As special relativity is a theory and not fact it is in a certain sense speculative (in that it has non discharged assumed premises within the conclusion of the argument) If the reality of time is speculation it is useful speculation that allows man to engage in art, math and science, whereas, a subjective universe (like the one you have suggested)is not a particularly useful theoretical construct. Also this view has considerable metaphysical problems. For instance how do you account for the homogenous quality of human experience that allows for standard observation in a universe that works as you have suggested?