This is correct, but based on specific assumptions. If the host flipped a coin and selected a door and 1/Nth of the time it had the car then you switching would not change your odds.

Similarly, an adversarial host who sometimes opens a door and sometimes just shows you your choice can similarly mess with the odds.

It’s really due to the host both being forced to show a door and knowing which door to choose that you gain from leveraging that knowledge.

No, if the host is acting randomly no information was revealed. It is equally likely that the car is behind your door or the other door.

The player are indifferent to switching under your scenario. Why not, I suppose?

That is also a very interesting way to think about it, if I understood you correctly. Seeing the host as another player, any "bad luck" he has, should translate into myself having a higher chance of success if used correctly.

With the host as another blind player, his opening of 98 goat doors only increases the probability you were right from 0.01 to 0.5, so still makes no difference for you to change. But of course the original version of the problem is predicated on the host knowing where the car is and only revealing goats.