Would matter falling into our black hole come shooting out of the white-hole we see?
Would time on either side of the event horizon even be related?
We would see that at the edge (if we could see it) there's new mass and energy, but that would be obscured by what appears as the CMB for us.
> Heck, Hawkins radiation means black holes can evaporate. Does that correspond to a universa collapsing?
No, it corresponds to stuff disappearing from our universe, until someday nothing is left.
> Would matter falling into our black hole come shooting out of the white-hole we see?
We'd see it as being part of the Big Bang, behind the CMB, so we wouldn't see it.
> Would time on either side of the event horizon even be related?
Er, not in any way that we could observe, so it's not a question we could answer. But we could receive information from the outside, except that the CMB would hide it from our prying eyes.
The idea that any mass that will ever fall into our black hole all simultaneously appeared at the big bang doesn't feel correct. That suggests a very specific relationship between time on the inside and outside. There are at least two moments that are distuinishable on both sides. The singularity appearing, and the singularity dissapearing. Compressing all time on the outside into the big bang means weird things for the timing of when the singularity dissapears.
It has to be the case that black hole evaporation means that mass-energy inside the black hole "disappears" from inside the black hole. It doesn't disappear from the rest of the universe, but if we are inside a black hole then we would "see" (if we could) that Hawking radiation means stuff disappears from our inside-a-black-hole-universe.