Do quest monsters follow the same rules for initiating attacks as regular monsters? I mean, can they only attack once per turn? What happens if they have more than one life and are wounded?
Also, can then initiate another attack in the same round if I pay a movement to repeat the quest?
yes, you are correct. Quest monsters follow the rules:
1 attack (they are attacking the hero)
Hero may spend a movement to attack after the Quest monster's attack is resolved
Place tokens on the Quest monster to indicate wounds it has if it has more than 1 Life