1.) Can you move after taking an action?
Not normally. The only exception is if you try to extinguish a fire, and fail. Then you take one movement to move your gnome out of the room to safety.
2.) Is it possible to: Move into a room with a fire. Attempt to fight fire and fail to put it out. Move out of room. Then do Faint check and update in a room next to the fire?
Yes, because of what I said for #1
3.)Can a fire start in a room with water (half or full)?
No, fires can never start in rooms with water.
4.) Can you enter a room with a fire with out Grog or extinguisher?