1) Re: Special encounters. Say you have the Yog Sothoth special encounter that you can draw when you encounter a gate. Normally you'd draw an other world encounter. Would the special encounter count as an other world encounter?
No, it counts as a Yog-Sothoth special encounter. The urban guide helps you, assuming the gate is on a city.
Same for things like Rumours that have their own test that replaces another type of encounter (ie: test obs instead of resolving an other world encounter)
I didn't quite follow your thread of thought. If you encounter a rumor, that's what you encounter. You don't do anything with gates, regular encounters, or expedition encounters unless, for some reason, the Rumor tells you to do so.
2) Re: Moving encounter tokens. Say you have an investigator A on the Amazon with the expedition token. Say another investigator B is on the the Himalayas. During the encounter phase, the Amazon expedition gets resolved and the next card is the Himalayas. Can both investigator A and B resolve expedition encounters?
Yes. It rarely happens because most people don't hike into the middle of nowhere for no reason at all, you have to get lucky to have the expedition move to where you are.