Do you fight first then get sucked in?

Or are you sucked in with no fight?

Does getting sucked in use a movement point?

If you move to an open gate during the movement phase you have to fight/evade the monsters. If you for some reason get there during another phase you don't have to fight (this turn) since fighting monsters on the board are done during the movement phase.

Entering gates are (mostly) done instead of having an encounter during the arkham encounter phase and it never costs you any movement since it's not done during the movement phase.