If your investigator is using his speed to go through Arkham and has to go THROUGH a spot that contains a monster, does he have to sneak and/or fight him, OR can you just go to the destination spot?

You always make the evade check if you're either leaving a space with or ending your movement in a space with a monster.

If you're using the Deputy Wagon, you only make the evade checks for your start and end points, since you "jump" to your new space without visiting any spaces between