\documentclass[addpoints,]{exam}

\usepackage{lipsum,graphicx,mdframed,array,ragged2e,booktabs,fmtcount}
\usepackage[table]{xcolor}

\pagestyle{headandfoot}

\def\arraystretch{3}

\newsavebox{\headbox}
\savebox{\headbox}{\def\strut{\vrule height\arraystretch\ht\strutbox
  depth\arraystretch\dp\strutbox width0pt}%
\parbox{\textwidth}{\hrule
  \strut\rlap{left}\hfill\mbox{center}\hfill\llap{right}\hrule
  \fboxsep=1pt
  \colorbox{gray!30}{\makebox[\dimexpr \textwidth-2\fboxsep][c]{\strut Breaking line\newline doesn't\\ work}}\hrule
  \strut number of questions is \numquestions{}.\hfill another one}}

\extraheadheight{\dimexpr \ht\headbox+\dp\headbox}

\firstpageheadrule
\firstpageheader%
{}%
{\usebox\headbox}%
{}

\begin{document}
    \begin{questions}
        \question some question
    \end{questions}
\end{document}