Is it legal for a US professor to require remuneration from the foreigner for consulting services? A person with allegedly academic affiliation from some foreign country approaches me with the request to help them in research being within my expertise.

The request anticipates about 10 hours of extra work without utilizing university equipment. The service can be referred to as consulting service.

Is it legal to require a personal monetary remuneration in this case from a foreign student/postdoc being consulted?

Alternatively, I can request being a coauthor in the future publication but I lack good reasons to believe that that person would not "forget" about this after they got a help.
It is not illegal to do legal work for money.  It may, however, fall outside of your institution's employment policies, and you might get yourself in a bind with your employer.

Consulting arrangements are generally laid out in your university's policies, and are often fine (but ask at your site!).  That said, I'll delve a little deeper into your question.  It is unusual for a "student or postdoc" to request the professional consulting services of faculty members, and thus you must consider what you are being asked to do.  If you are being asked to help the student cheat -- i.e., the student is planning on representing your work as something otherwise, you should take absolutely no part in it.  If it's something central to the students project, and you are being brought in as a contractor and not a collaborator, you might consider avoiding it.

The fact that the party is foreign is not germane.