i just had an encounter which opened a gate at the location and a monster appeared.
1. Does the investigator immediately have to fight the monster?
No the investigator is sucked through the gate before he needs to deal with the monster.
2. Is the investigator sucked through the gate? Because i can find this rule only at the mythos phase. Does it also apply here?
Yes, it applies here as well.
3. But if he is sucked through, what is happening with the monster. Because there is a rule which states, that monsters from encounters are disappearing after the encounter. So it would be pretty useless having the investigator sucked through the gate and the monster which just appeared is immediately disappearing.
The monster stays as a little present for those remaining behind in Arkham.
Only when the encounter says "a monster appears" do you deal with him right now. If you manage to escape him or get defeated by him, then he goes back to the monster cup.
When a gate appears with the monster, it all gets changed up. You are drawn through and delayed. The monster stays there (and will move if his symbol comes up on the mythos card).