This is great! And as it should be. Definitely not the case in China, and there are a lot of stray cats in cities these days - I've thankfully not seen any accidents but I'm sure the street sweepers would tell a very different tale.
As it should. Not sure about the rules in Taiwan, but I think it's a cultural thing to tap the car bonnet loudly before getting into the car, especially in winter times, so any animals that might be sleeping under the car will know to leave.
You fail your driving test in Japan if you don't check under the car for animals before starting the engine.