Suppose a Russian citizen who studies (hence lives) in the US [on an F-1 visa] wants to get a Schengen visa to visit the Netherlands. If they happen to be in Russia during a break, can they apply for such a visa through the Netherlands Visa Application Centre in Moscow, or will their application be denied (asking them to apply for a visa in the US where they currently reside) since they do not currently live in Russia? Further, whatever the answer to this question is, does it generalize to visas of all other countries, or is it specific to the Netherlands (or Schengen visas)?
You can request Schengen visa from other countries than your residence.
That might require explaining why it is a special hardship for you to follow the usual process, and starting a visa appication by explaining why you are a special case is not the best way to go about it. Officials like orderly, standard procedures.
Check if it is still possible for you to follow the usual procedure. If so, do it.
You cannot generalize.
Most countries reserve the right to issue visas to anyone, anywhere, if they feel like it. But getting into that special process may be difficult.