I'm not sure if this has come up before. Let us suppose Way of the Mouse has Fishing Village. Here goes.
We know this:
Throne Room -> Wharf / Way of the Otter. Throne played a Duration, is discarded from play when the Duration is. Wharf is discarded from play this turn; so is Throne Room.
That bit is ambiguous. You Throne Roomed a Wharf, so you play Wharf twice. Playing it using Way of the Otter or "normally" is independent for both plays. Your outcome is true only if you chose to play Wharf using Way of the Otter
both times. If you play Wharf at least once "normally", then Wharf will have something to do on your next turn, so stays in play until then. Since the Throne Room played the Wharf, a Duration card, twice, it will also stay in play until then.
Okay, so here's the question:
Throne Room -> Necropolis / Way of the Mouse -> Fishing Village. Necropolis played a Duration card which acts next turn; Necropolis stays in play. Throne did not play a Duration card, is discarded this turn.
Correct.
Throne Room -> Wharf / Way of the Mouse -> Fishing Village. Wharf played a Duration card which acts next turn; Wharf stays in play....
Well, Wharf played a card that was not in play "leaving it there", that's what's keeping it out until the card it played would have stayed out. That it played a
Duration card is not relevant, Duration cards are just the overwhelming majority of cards that can stay out.
In this case, there actually is no existing card that can be a card for Way of the Mouse and either is like a Throne Room or like an Overlord, it could only be a Duration card that would stay out. But that is AFAIK a "lucky coincidence" of how cards are priced (those effects don't come so cheap that a card is eliigable for Way of the Mouse).
In your previous Overlord -> Wharf example, it could have been Overlord --> Throne Room --> Wharf. Throne Room would have stayed in play along with the Wharf, so Overlord actually stays in play as a stand in, as long as Throne Room would, which is as long as Wharf does. So Overlord played a non-Duration card that would have stayed in play, causing Overlord to stay in play.
...Throne did play a Duration card (Wharf), stays in play until Wharf is discarded, next turn.
It may be obvious, but it's worth repeating that the important part is that Throne Room played a staying-out Duration card
twice. If it was a Vassal or Herold that played Wharf, either normally or using Way of the Mouse playing Fishing Village, Vassal/Herold would not stay out.
In other words, Throne on a Duration card played according to a durable Way of the Mouse behaves differently than Throne on a non-Duration card played according to a durable Way of the Mouse.
Yes. That's because the rules for "Throne Rooms" staying out explicitly mention they only apply when they play a Duration card extra times. It may seem strange to make a distinction, but having it work on all cards would have resulted in another change in what stays out for "Throne Room -> Throne Room -> Smithy/Fishing Village", keeping the first Throne Room in play "again". Basically one aspect of the 2019 errata was to keep the rules changed to a minimum, with getting the work done on errataing Overlord/Band of Misfits a.s.o to use the new "play something, leaving it there" method and keep tracking of effects half-way intact.