What happens if they deal damage to each other, for example during combat? Will the vampire die or not? I think that the vampire will die since at the time he would receive the +1/+1 counter for killing the wolf he is already in the graveyard. Is that right?
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Yes, that's right.
As the Wolf dies the Vampire's ability will trigger and be placed on the stack. Unfortunately, by the time you get round to resolving it, the Vampire will be in the graveyard and unable to have a +1/+1 counter placed on it.
That is correct, the vampire will die. Combat damage is dealt simultaneously if none/all of the creatures have First Strike/Double Strike (if two creatures were to fight Contested Cliffs, the situation would be the same). After damage is assigned, it is dealt simultaneously, then triggered abilities are placed on the stack. This happens just before a player receives priority, which means that you have to check for state-based actions, one of which is to place creatures with lethal damage in the graveyard.