After arriving in the United States with a U.S. tourist visa, I directly go to a lawyer to apply for refugee status. Can I stay in the United States legally for a long time?

Theoretically it is possible (legally possible), but in practice it is very difficult. Not everyone can apply for a green card, and not just those who want to apply for a green card can enter the green card application process. Law firms have business rules and regulations, and you can't mess around. Unless you have irresistible factors, such as being threatened by jail, death, persecution, etc., you may be asking for trouble. The Immigration Bureau will consider that you deliberately stayed and failed to return, leading to deportation. This situation has been reported frequently over the past dozen years. There are not just one or two deportees.

It is recommended not to take this risk.