How often do Japanese use soy sauce?

All the time. Japanese keep it on the table as westerners do salt and pepper. And they reach for it automatically when the food is served. It's an essential part of a Japanese meal - as basic as rice. Many people use it on western food too. There are jokes about Japanese who carry their own soy sauce in pocket containers when traveling abroad.

How do they use it?

At the table, in two basic ways. They might pour it directly onto the food, like pickles for example. Or they might pour some into a tiny sauce dish and dip pieces of food into it.

Do they every cook with soy sauce?

Yes, shoyu as Japanese call it, is a basic ingredient in many dishes. Suiyaki, yakitori, and teriyaki are some of the dishes that use shoyu in the cooking or preparation. And shoyu can be used as one of the ingredients for marinating before cooking.