Does analysis class diagram contain a menu and interface?

I am implementing a class diagram for an application related to teachers and students as an assignment. A student can have many teachers.

I will implement a menu in coding. Should I add menu and interface as classes?
If yes, how should I link all these?
What can menu and interface contain?
Could I say: menu has settings and teachers page, which has all teachers profiles in it?

A class diagram with only classes belonging to the domain of application (e.g teachers, students, classes, ..) without any of the app's internals (e.g. UI related classes) is called a "domain model". Its goal is not to document all possible classes used in your app, but to focus on the domain knowledge, independently of how the app is structured or implemented. Such domain model would stay the same if you create a mobile app, a text based terminal application, or a voice controlled app.

On the other side you may also want to give insights on how your app is designed, i.e. using menu classes, database connectors, etc... This is also fine. It's up to you to decide what is the purpose of your class diagram and what you want to show with it. The only thing that you should avoid is to have one huge diagram with everything, because it'll be too complex to explain anything. Better go with several class diagrams focusing on different aspects of your design.

One thing that may help if you want to combine in the same diagram classes with very different purpose, i.e UI classes, domain classes and business logic, is the Entity-Boundary-Control stereotypes, to help to understand what class fulfills which technical purpose.

Edit: Analysis diagrams

Your title and your comments refer specifically to analysis diagrams. Analysis diagrams are not defined by the UML standard and depend on methodological choice:

• Some methods see analysis models as self-standing with only elements of the problem space, whereas design diagrams show the solution space.

• But some other methods see analysis models just a first stage that evolves into a design model.

• The Unified Process defines analysis classes to be Entity Boundary Controls, because they are directly related to Use-Cases, which model requirements. They would not contain details such as menus (which will belong to refinements of the boundary classes). But they would contain more than only entities of the domain model.