A Schwarzschild white hole is a time reversed black hole. On the inside it has a singularity that explodes with energy and matter at the moment when the internal time starts. After this initial moment the singularity no longer exists on the internal time scale. The internal time of the while hole is disconnected from the external time by the event horizon. Therefore, the question, "When did the singularity explode?" has no meaning for an external observer.
Matter and energy created in the explosion crosses the event horizon from inside out and flies away at different moments of outside time. With matter and radiation leaving, the mass of the white hole becomes smaller, its event horizon shrinks until eventually disappears, and the white hole ceases to exist.
A white hole does not repulse matter. Gravity is always attractive. On the inside, the direction of time is radial from the singularity to the event horizon, so matter simply is moving in time. Passed the horizon the radial coordinate becomes spatial on the outside. Stuff is brown out through the horizon at the speed of light (which is relative) and flies away on inertia.
Aside from the initial singularity, which is expected to be described by the future theory of quantum gravity, there is no exotic matter inside or outside the white hole. It's event horizon cannot be crossed from outside to inside. So researches on the inside cannot receive any information from outside and have no way of knowing in what universe they will end up when they are thrown out through the horizon. It is believed that white holes do not exist, because they would violate energy conservation by producing matter from nothing.
Expert comments are welcome and I would edit the answer accordingly, if any part if it is incorrect.
