2. The major of pure mathematics is mathematics.
3. The actuarial science you mentioned is economics, and it is only an interdisciplinary subject of economics and mathematics. Moreover, partial economics. In American universities, these departments are all economics departments.
If you want to study pure mathematics, and your future goal is to do research and become a university professor, you need to study until you get a doctorate. If you want to go back to China to find a job, no matter what you want to do in the future, you should at least go to graduate school (because all the returnees in the United States now have graduate degrees or above). If you want to stay in the United States, you can try to find a job after graduation.
The foundation of mathematics in the United States is very simple, but inexplicably, the difficulty, intensity, depth and speed of advanced mathematics are faster than those in China. If it is only high school, high school mathematics in the United States is junior high school mathematics in China. Mathematics in universities is more difficult than in China.
6. Yes, there are fewer girls studying science in America.
7. If the major is pure mathematics, the counterpart's job is to be a teacher, a professor and engage in research. Being an actuary is wrong. Teachers in compulsory education are more difficult for international students. If you want to stay as a professor in a university, you need to study all the time, finish your postdoctoral studies, and then stay in school to do research before you have the opportunity to become a university professor. So can an actuary. I heard that the exam is very difficult and abnormal.
If you are good at math and have logical thinking, you can actually try to study some science and engineering departments such as CS and EE, which is better than the math department in finding a job.