Is adjunct faculty's duty limited to teaching? Do they have any role in course design, grading, etc.?
What are the other responsibilities, as an adjunct faculty?
There's a wide range of people called adjunct faculty. For example, it may include working professionals in the field who serve as part-time faculty (this is common in law and architecture, and I've seen it with industrial researchers in computer science). Depending on the circumstances, this sort of adjunct may be treated much like the other faculty in the department, except for being part time. However, I assume you are talking about the most common use of the term "adjunct", at least in the US, namely low-paid, low-status faculty typically teaching part time on temporary contracts. (They may be working full time overall, but only part time at any given institution, so the universities can avoid paying benefits.)
For this sort of adjunct, it really depends on the details of the contract, and it may vary between universities, as well as depending on issues such as whether any of the courses are online. A typical arrangement will include some amount of course design (at least at the level of creating a syllabus), lecturing, grading, and office hours. Typically there is no committee work or other service and no research duties.