If I end my turn on a gate and get sucked into one of the other worlds, do I have an other world encounter on the same turn? Or would I just have an encounter on the second, right-hand space after my move to that?
The Other World Encounters Phase occurs after the Arkham Encounters Phase, so you do in fact need to have an encounter on the first space of the Other World the same turn that you move onto it.
Also, I'm working on the assumption that flying monsters can only directly move to a street space 1 movement space away (i.e. streets are not interconnected across the board) for the purpose of their initial movement. Is this correct?
Flying monsters can only move one space at a time, yes. But if they cannot reach an investigator in 1 move, they instead move to the "Sky". From here, they can reach any street on the board.