As a student with questions about a paper, can I email the author myself or should I ask someone else to contact him?
You can definitely contact a paper author. They might be of the 'it obviously follows' == after 5 pages of calculations kind, or the empiricist who published the 20 successful regressions or simulations out of 200, with 180 contradicting their result or being inconclusive; and in either case ignore your question. From personal experience though, it can even lead to breakthroughs: in my case, someone sent me his lecture notes which clarified something I was stuck with, and related to the submitted question. However, if your advisor knows the author, or simply is well-known in their field, do mention that you are their student, as it should increase good will on the author's part - after confirming with your advisor that they are cool with it. Showing that you are active, interested, and independent should also go down well with the advisor.