Is it possible to temporally make undefined \csname control sequence not silently expand to \relax?

You cannot alter the behaviour of \csname, but you can use e-TeX's \ifcsname to issue an error:

\csname
  \ifcsname#1\endcsname
  \else
    \expandafter\UndefinedName
  \fi
  #1%
\endcsname

where \UndefinedName is deliberately not defined. TeX will then issue an error complaining about Undefined control sequence, followed by #1 (which is what you actually want to complain about).

(Unlike \@ifundefined, \ifcsname is expandable and does not add to the hash table.)

You can check the command for existence and invoke the error yourself.

\@ifundefined{/#1/}{%
  \GenericError{}{%
    Command /#1/ undefined!
  }{%
    See the definition of \protect\mymacro!
  }
}{}%

The \@ifundefined macro uses \csname so it will define the macro as \relax. I don't think that this is an issue (we invoke the error anyways), but you can surround the code by \begingroup ... \endgroup to make the definition local.