1. There is no problem in class at all, but if you want to have a class, you really need to find out where that class is. The time and place of classes in that building are posted on the windows on the first floor of each teaching building, or you can check on the website of Fudan University Academic Affairs Office, and there is a course selection guide (that is, the curriculum schedule).
In the library, students from other universities can register temporarily. If you are not a college student, you should be ok. You can ask the front desk and bring your identification.
There is no specific study room, that is, there is no classroom for classes. Similarly, if you find a classroom without classes in the window posted on the first floor of each teaching building, you can go in and study for yourself. Generally, it will close at night 10: 00, but classroom 3 108 is an all-night study room.
(In addition: the first number of classrooms in our school is the building number, the second number is the floor number, and the third and fourth numbers are the classroom numbers. For example, 3 108 is the third teaching building, the first floor and the eighth classroom).