Yes, you can use this. In that context, you are putting the "student" in the category "beginner". It would be no different from saying "computer screen" to describe a screen that works with a computer.

"Beginner" is a noun... usually. When it is coupled with the word "student" (also a noun, but not the only noun that could be put there), the nature of the word changes to an adjective, even though its meaning doesn't really change at all. This is generally acceptable, unless the adjective-noun pairing contradict one another, such as a novice expert (a novice has little knowledge of a subject, while an expert has a lot of knowledge of a subject.)

Yes. It is very common to use nouns as adjectives in English. The noun "beginner" is used in as an adjective in the name of the series "Beginner Books," a famous series created by Dr. Seuss. We understand that "beginner books" means "books for beginners." There isn't really any rule of grammar. The rule is "you can do it if you think it will make sense to the listener." The meaning is often logically ambiguous, but understood from context. For example, the "American Dove Association" supports birdwatchers who are interested in doves, but the "American Cancer Association" is not an organization that supports cancer! You can stack more than one noun. For example a "cathode ray tube oscilloscope" is an oscilloscope that uses a "cathode ray tube," which is a tube that creates "cathode rays," which are rays that come from a cathode!

Yes. But not always, I guess it depends on the case.