Try using this for your style document
\documentclass[12pt, a4paper,twoside, openany]{memoir}
the default option is openright instead of openany, and that will remove your blank pages.
EDIT :
I deleted \usepackage{frbib} and replaced your last lines by
\usepackage{lipsum} % for example
\begin{document}
\section{Section title}
\lipsum
% *************** Fin du style ***************
\frontmatter
%\input{front.tex} % I don't have this file
\mainmatter
\begin{otherlanguage}{english} % Added on 17 April, 2014.
%\input{introduction/introduction.tex}
%\input{pubs/pubs.tex} % Publications
% bibliographie
\bibliographystyle{unsrt}
{\small\bibliography{bibliographie_these}}
% *************** Appendices ***************
\appendix
\appendixpage
\section{Appendix section title}
\lipsum
\end{otherlanguage}
% *************** Back matter ***************
%
\backmatter
%\input{back.tex}
\end{document}
It works just fine, so your problem should come from \input{app1/app1.tex}.
\appendix, insert\cleardoublepage \let\cleardoublepage\clearpage. this should prevent all blank pages thereafter, so if you want a blank page before (say) an index, what is needed would be more elaborate. – barbara beeton May 21 '15 at 12:56