Is it possible? Can a figure move to a space with another figure and spend an action to open a door, for example?

Could Gaarkhan move an spend an action of charge in the same case?

No, this id generally not allowed. Check the Rules Reference Guide under movement. There it says that you can not end your movement in a space containing a figure. There was a clarification fro. Paul or Justin stating that this means figures can not interact with objects from spaces occupied by figures.