No, time travel is not allowed by the laws of physics. Yes, we are all travelling forward in time at a rate governed by the gravitational field in which we are immersed. Time on a big planet goes ever so slightly slower than it does on a planet with weak gravity. Time travel into the past is definately impossible by even the most conjectural theories of multiple universe quantum mechanics. In fact, quantum mechanics says nothing at all about time travel. Time is taken as a dimension of a physical system that determines things like the flow of energy and entropy and causality is determined on a strictly probabilistic level for microscopic objects.
Perhaps the easiest way to see the impossibility of time travel is to look at the paradox of time travel. If you could go back into the past you could shoot yourself, thus, time travel would make it impossible to ascribe a direct causality to any particular event. This is a literary arguement, but it is still logically sound. No method of sending even an energy signal into the past has ever been observed by science. The entire question is wrong to ask of science which observes and describes the universe as it actually exists.
A final arguement against time travel comes from thermodynamics. In any system travelling backwards through time entropy would be decreasing. This is never observed. Two objects in contact at the same temperature do not suddenly start heating and cooling respectively as they might if time flowed backwards. Entropy has been called the arrow of time, as it points the direction in which energy flows in any thermodynamic system.
Modern scientific theory is filled with fantasy ideas from science fiction, fantasy, and just plain speculation. Black holes, white holes, wormholes, strings, and multiple universe interpretations of quantum mechanics have all been postulated as possible mechanisms of time travel at one time or another. Such exotic theoretical constructs are not only unobserved directly in the universe, they are by definition onobservable, which makes any use of them in theories of time travel quite convenient as the actual properties of the observable universe have no part in the purported mechanism. Science is concerned with the universe as it is, not as how we would like it to be. Time travel is so much wishful thinking that has nothing to do with the world we actually live in.