White box tests evaluate the behavior of a feature. Whether a given input will produce the expected output or whether code has passed the desired flow. To run this type of test you need knowledge of the source code (other than a black box test where only the database is evaluated).
1. How does the white box test run?
The most formal way to do this is to use a unit test battery. It is also possible to simplify by separating a portion of the code and creating a new project / file for this purpose.
Is this test related to only methods of a given class?
No, the test can check whether a particular part has been executed or not, but can also test an entire method.
Some unit testing tools provide diagnostics based on test cases and say which lines / snippets of code have not been run, which serves as an indication that not all paths have been executed or there is a dead code.
What types of methods should I submit to the white box test?
Any type especially has some validation or rule rule application.