At what stage does combat actually start?
Assuming you ask about 3rd edition rules. From rules reference:


  If a hunter is on Dracula’s current location during dawn or dusk, he
  must start a combat with Dracula.


And also:


  Dawn occurs before day, and dusk occurs before night.


So, combat with Dracula only happens between action phases, not immediately.

EDIT:
To clarify your scenarious:

Scenario 1: Other Hunters continue to do their actions for this phase, then combat occurs. Since movement normally only allowed during the Day, I assume combat would happen at Dusk, and night effects on combat cards would apply.

Scenario 2: Since it's just beginning of the Night, it means that combat at Dusk has just concluded (for described situation to arise Dracula already was in the city when Hunters moved in during the Day). So all Hunters carry on their actions. THEN it's Dracula's move, and if Dracula stays in the same city with Hunters or moves to city with another Hunter, combat at Dawn occurs.