If you fall into a gate during an encounter in Arkham, you have an encounter in the first area of the Other world in the 'Other world encounters' phase' during the following turn, not the same one. So, the number of encounters is correct, but the sequence is split in three parts.
Not so sure of your intent on this one. If you voluntarily enter an already-open gate during the Arkham Encounters phase, you should wind up with only 2 Other World encounters before you return.
If, however, you get an Arkham encounter that says "A gate and monster appears!" then you will end up with 3 OW encounters.
This is why:
• Turn 1, Phase 3 (Arkham Encounters):
"A gate an monster appear!" you are drawn in to the OW and delayed. Being delayed when you enter the OW is ultimately what adds this extra encounter.
• Turn 1, Phase 4 (Other World Encounters):
You have your first encounter in your other world.
• Turn 2, Phase 2 (Movement):
Your character is delayed, so his entire movement is to stand upright. But he's still in the first OW space.
• Turn 2, Phase 4 (OW Encounters):
You have your second encounter in your other world.
• Turn 3, Phase 2 (Movement):
Your character is not delayed, so he moves to the second OW space.
• Turn 3, Phase 4 (OW Encounters):
You have your third OW encounter.
• Turn 4, Phase 2 (movement):
You are in the second space of the OW, so you return to Arkham.
If you had entered the OW voluntarily—i.e. there was already an open gate there and you ended your movement on it, then you would not be delayed when you entered the OW. Therefore, you would expect to have 2 encounters before returning.
If you were drawn through a gate during the Mythos phase, you would be delayed, but you would end up getting 2 OW encounters because the next movement phase would come and un-delay you before you got your first OW encounter.
Of course, the number of turns you are in an OW can be affected by if you get delayed or find a quick way back, or if you have the Find Gate spell.