How do white holes form? Why are they called the "time-reversed" versions of a black hole? What's an "eternal" black hole? How come a white hole is only encountered if you remove the star from the space-time surrounding the black hole? Why is it that if you add any matter to the area that the white hole disappears? Any explanation greatly appreciated.
How do white holes form?
No physical process which would form a white hole is known.
What's an "eternal" black hole?
This is exactly the only speculated way a white hole could exist and not violate known physical laws - it would have to be here "since always" with it's eternal black hole counterpart. Because it cannot form dynamically, there would have to be no physical time when the black/white hole pair did not exist.
But even if an eternal white hole did exist, it is not entirely clear how it would look like, because it could not be accessed from outside and it would only spit everything from it's own interior. But what should we expect from it's interior?
Why are they called the "time-reversed" versions of a black hole?
Equations of fundamental physics are usually time-reversible (particle physics is time-reversible in a CP or CPT sense, though) which means you can take a certain reversal of the solution of your equations to obtain another solution. A white hole is a complementary object to a black hole in this sense. Every particle in it's field behaves the same as a particle near a black hole only with a reversed arrow of time.
How come a white hole is only encountered if you remove the star from the space-time surrounding the black hole? Why is it that if you add any matter to the area that the white hole disappears?
This is a remark upon how you can formally obtain the white hole when sitting at a desk and solving Einstein equations. It does not describe any kind of physical process.
It expresses the fact that you can find a perfectly time-symmetric solution of Einstein equations which has an eternally static black hole and it's counterpart, the white hole. However, the solution of Einstein's equations involves all of space and most importantly all of time. When you take your solution and try to figure out a modified space-time where a star lives a happy life and eventually collapses onto a black hole, you find out that the patch of the space-time which could be originally continued into a white hole has to be terminated in the interior of the star.