Tsunami's are mostly caused by earthquakes under the sea. A lot of animals can sense these earthquakes before the waves arive, and run for the hills.
Earthquakes don't just happen all of a sudden, there's some minor shifting and shaking in the earth before the earthquake. Maybe animals can sense these and base their course of action on that.
About weather and animals: Thunder and stuff is pretty predictable, the air gets tense, humid and warm. Extreme weather alters properties of the air before it hits. So birds can sense those changes and get into safer conditions.
About the horse story: I have no clue why the horses ran to the corner. I think it's probably possible to find out, but then we'd need to know why the stables collapsed. If it was due snow on the roof, the may have gotten nervous because of the slightly bending structure. If it collapsed because of something else, other things may have alerted them.
Animal have far better senses than humans, so this is only natural.