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For computer science (CS) courses at Harvard, a course’s penultimate (i.e., second-to-last) digit ordinarily signifies its “course grouping” (i.e., area within CS), per the Harvard College Handbook for Students:2: Theory3: Economics and Computation4: Hardware and Networks5: Programming Languages6: Systems7: Graphics, Visualization, and User Interfaces8: Artificial IntelligenceAccordingly, CS50 (whose penultimate digit is 5) falls under the “Programming Languages” heading, as it’s a software-centric course. Meanwhile, for undergraduates concentrating (i.e., majoring) in CS, courses numbered < 100 are primarily for undergraduates, courses numbered >= 100 and < 200 are primarily for undergraduates and graduate students, and courses numbered >= 200 are primarily for graduate students. Within the concentration (i.e., major), CS50 is indeed primarily for undergraduates. Though a non-trivial number of graduate students from fields outside of CS also take the course.

In the US, professors are expected to provide all students with a uniform course syllabus at the very start of the course and adhere to it during the course. This generally includes information on all course assignments and exams, how they will be graded, and how those grades will be combined to yield an overall grade in the course. The faculty handbook for the institution spells out the details of what a syllabus is supposed to include, at a minimum.For the sake of fairness, the syllabus is the same for all students in the course. Some professors may offer the opportunity for students to do additional optional challenging assignments for extra credit; that opportunity is open to all students and is part of the syllabus. Occasionally there are situations where a senior undergraduate course and a beginning graduate course with overlapping material will meet together in one classroom. In that case the syllabus, assignments, and expectations for the undergraduates and the graduate students may differ.So the answer to the OP’s question is no. In the US, college professors cannot arbitrarily assign harder homework to students they wish to weed out of the class.

Yes. For details, see the relevant entry in the Handbook for Students.Harvard hosts the Harvard-Smithsonian Center for Astrophysics, one of the largest astrophysics institutes in the world. But I notice that the astrophysics track attracts relatively few undergraduates, probably because many students interested in the field prefer to do physics, which has a lot of overlap in terms of coursework and is somewhat broader, offering more flexibility post-college.