Do monster encounters only occur during the movement phase when leaving a space with a monster or ending at one, and also when cards say a monster appears? Flying monsters moving to your space don't induce an encounter right?
If you start your movement phase on a location or Street with any monsters, you must either evade them all or fight them.
If you attempt to pass through a location or street with a monster on it, you must either evade or fight them.
If you end your movement on a location or street with a monster, you must either evade or fight them.
Flying monsters usually move during the mythos phase, so you won't need to fight them till your next movement phase.
Monsters that appear due to encounters need to be fought or evaded immediately.
An exception to the above is when you return from an other world location and have an explored marker. On your first turn after returning, you don't need to fight or evade any monsters guarding the gate, giving you one relatively safe shot at closing the gate
For the exception above, you can find it in the rules on page 18
During the turn he returns to Arkham from an Other World, an investigator does not have to evade or fight any monsters in the gate’s location. This rule only applies during the turn he returns to Arkham; in subsequent turns, if the investigator remains in the location, he must evade or fight any monsters there as normal.