How can I add the bibliography in a report with its own section, including number?

Preliminary remark: In the report class, the bibliography will by default be typeset as an unnumbered chapter. If I understand correctly, you do not only want the bibliography to be numbered, but also "downgraded" to a section.

One solution is to selectively change the definition of \thebibliography. Add the following to your preamble:

\usepackage{etoolbox}
\makeatletter
\patchcmd{\thebibliography}{%
  \chapter*{\bibname}\@mkboth{\MakeUppercase\bibname}{\MakeUppercase\bibname}}{%
  \section{References}}{}{}
\makeatother

In the main document body, just use

\bibliographystyle{plain}
\bibliography{bibliography}

I left out \cleardoublepage for the sake of consistency (sections do not automatically start a new page).