Is the keeper able to spread fire through ladders? The card says adjectant room, and usually the two rooms connected by a ladder are not adjectant. Then again, its possible to move between both rooms as if they were. Same question could be asked for vents.

Quoting the rules:

Ladder Feature - Investigators and humanoid (blue) monsters may move between spaces containing ladder markers as if they were adjacent. The spaces are considered adjacent for movement purpose only. Other monster types may not move using ladders.

Don't think the fire can spread through ladder markers, IMO.