What I believe:
• The Mirror Universe did not exist until the ion storm and transporter beam intersected to create it. The MU was a "flash copy" that just popped into existence. Transporters work on subspace carrier waves, which can bleed into other dimensions.
• The Mirror Universe was an inaccurate copy, like a bad OCR document full of misread characters. That's why it was "evil." Also, all of its history and memories were new and illusory— they were skewed, inaccurate copies of what was known and remembered in the real universe.
• The Mirror Universe was not a whole universe, just a local bubble encompassing the Halkan planet, the Enterprise, and the illusion of stars in the cosmic distance. And the whole bubble could be similar to a holodeck simulation, with no physical substance at all, but created as an accidental phenomenon.
• The landing party traded minds with their "evil" counterparts; while the bodies (or the mere image of MU bodies) of each group stayed in their own dimensions.
• After the landing party's minds are freed from the MU bubble, the bubble ceases to exist; it wears out and expires. If the group had not escaped, their real minds would have ceased to exist, and the "evil" group in our universe would, in essence, be like brain-damaged versions of their real selves.
• The spinoff visits to the MU, that I recall in DS9 and ENT, are not "true stories" in the TOS universe. They would be considered enjoyable science fiction if TOS is fact. Like, Mr. Chekov streams ENT on SpaceFlix and thinks its a hoot.