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.

Definitely contact the author. Collaboration is what research is all about. Authors expect these sort of emails when they publish. Also, sending email like this lets people know your name, one person at a time. This way, when you're at a conference later on, you can go over to the author and say, "hey, I emailed you a while back, nice to meet you in person." It's always good to network.

It would be good form to mention your advisor in the email, whether he's well-known or not.

I will give the point of view from Eastern Asian Universities.

Here the Lab culture is too focused on the professor as the head and only public face of the laboratory.

Because of this, many students are not used to being asked directly about their research, and usually they do not know what to do about it, and will end up asking their professor.

The best case scenario is that the professor won't mind and will give the student authorization to mail you back, but the worst case scenario (it happens!) is that the professor will get offended because you contacted a student and not him, and you won't get any answer at all.

This mostly applies to Universities in China, Japan, Korea, etc. I would recommend mailing the professor and asking him directly, it will take time, but is usually your best bet. Even more, it is way better if you get your adviser to contact him for you, and then he can ask for his permission for you to contact the student directly (I'm really not joking about this)

Unless the guy writing the paper is foreigner, then go ahead.