i played with Dreamwalker hero (he have the box item - which allows him in any faze : instead of moving to archam move to any open gate location) and i was first player i drawed mytos card - move first player to strange house in the hill.
Whats correct :

1) He is moved to strange house in the hill

2) Any faze : use Box item and be moved to location with opened gate

Anarin said:

#1. Gate Box only works when you "return to Arkham".

yes but strange house in the hill is archam

But "Traveller Lost in Fog!" Mythos doesn't use the phrase "return to Arkham", it says move FP to SHHitM.

An effect that has you go to Arkham from an Otherworld is different from "Return to Arkham" game-wise. A couple of encounters back us up with this. There's an Abyss(?) encounter that "moves" you to the Black Cave. Even if a gate is open there to the Abyss, you don't get an explored token. Why? Because you didn't "Return to Arkham," you simply "Moved to the Black Cave."

Strange, maybe, but understandable if you want to get to the nitty-gritty of it all!

Yup, I concur with Ecno and Dam. You don't get an explored token if you are whisked off into a different destination other than "Return to Arkham". Willaim Bain can really ruin your plans that way.

The mists rolled into the Dreamlands, and suddenly you could not see a thing. You came upon a cabin while blindly groping around in the mist, and when the mists parted, you discovered you were in Kingsport.

You don't receive an Explored token because:

1. You're not on a gate in the first place
2. You have not "returned to Arkham"; rather, you were inadvertently spat out in an unpredictable manner. Even if for some reason there was a gate at the Strange High House, you still wouldn't receive an Explored token.

In the end it's not so bad. When Luke winds his way down into Kingsport, he can have a few encounters to stave off the rifts, and then take the train back into Arkham, enter a gate, and get the remaining clues required to seal.