A. The wet towel you get at restaurants in Japan, usually with a glass of water or a cup of tea, is called “Oshibori”. You can use it to clean your hands before you eat and you can also use it just like a napkin during a meal.
It’s usually chilled in summer and heated up in winter but some restaurants give you either hot or cold all year round. Which one they give you is all up to their preferences, no specific reason ! Sometimes it’s simply because they only have a machine to heat it up.