After hours of debugging, I finally found out that the header of my bibliography's last page is changed by using spacing in the text. Now my world is turned upside down.
Please have a look at this correct example. Both \begin{spacing} and \end{spacing} are commented out:
\documentclass[12pt, a4paper, oneside, onecolumn]{book}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[ngerman]{babel}
\usepackage{setspace}
\usepackage[round]{natbib}
\usepackage{fancyhdr}
\pagestyle{fancy}
\renewcommand{\chaptermark}[1]{%
\markboth{#1}{}}
\renewcommand{\sectionmark}[1]{%
\markright{\thesection\ #1}}
\fancyhf{} % delete current header and footer
\fancyhead[LE,RO]{\bfseries\thepage}
\fancyhead[LO]{\bfseries\rightmark}
\fancyhead[RE]{\bfseries\leftmark}
\renewcommand{\headrulewidth}{0.5pt}
\renewcommand{\footrulewidth}{0pt}
\addtolength{\headheight}{0.5pt} % space for the rule
\newcommand{\plainplain}{%
\fancypagestyle{plain}{%
\renewcommand{\headrulewidth}{0pt}
\fancyhead[LO]{}
\fancyhead[RE]{}
}}
\newcommand{\myplain}{%
\fancypagestyle{plain}{%
\fancyhead[LO]{\bfseries\rightmark}
\fancyhead[RE]{\bfseries\leftmark}
\renewcommand{\headrulewidth}{0.5pt}
}}
\plainplain
\pagestyle{plain}
\usepackage{lipsum, filecontents}
\begin{document}
\frontmatter
\pagenumbering{Roman}
\tableofcontents
\clearpage
\listoffigures
\clearpage
\listoftables
\clearpage
\mainmatter % implies arabic pagenumbering
\myplain
\pagestyle{plain}
% \begin{spacing}{1.5}
\chapter{Introduction}
\lipsum[1-7]
\section{New Section}
\lipsum[8-14]
\clearpage
\plainplain
\pagestyle{plain}
\backmatter
\pagenumbering{roman}
\setcounter{page}{5}
\nocite{mybook1}
\nocite{mybook2}
\nocite{mybook3}
\nocite{mybook5}
\nocite{mybook6}
\nocite{mybook7}
\nocite{mybook8}
\nocite{mybook10}
\bibliographystyle{plainnat}
\begin{filecontents}{bibliography.bib}
@book{mybook1,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
@book{mybook2,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
@book{mybook3,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
@book{mybook4,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
@book{mybook5,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
@book{mybook6,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
@book{mybook7,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
@book{mybook8,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
@book{mybook9,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
@book{mybook10,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
\end{filecontents}
\bibliography{bibliography}
% \end{spacing}
\end{document}
Note that there is only a small roman page number on the top right; the same applies to the previous bibliography pages.

This is the erroneous version where I use spacing; there are just two lines more (\begin{spacing} and \end{spacing}):
\documentclass[12pt, a4paper, oneside, onecolumn]{book}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[ngerman]{babel}
\usepackage{setspace}
\usepackage[round]{natbib}
\usepackage{fancyhdr}
\pagestyle{fancy}
\renewcommand{\chaptermark}[1]{%
\markboth{#1}{}}
\renewcommand{\sectionmark}[1]{%
\markright{\thesection\ #1}}
\fancyhf{} % delete current header and footer
\fancyhead[LE,RO]{\bfseries\thepage}
\fancyhead[LO]{\bfseries\rightmark}
\fancyhead[RE]{\bfseries\leftmark}
\renewcommand{\headrulewidth}{0.5pt}
\renewcommand{\footrulewidth}{0pt}
\addtolength{\headheight}{0.5pt} % space for the rule
\newcommand{\plainplain}{%
\fancypagestyle{plain}{%
\renewcommand{\headrulewidth}{0pt}
\fancyhead[LO]{}
\fancyhead[RE]{}
}}
\newcommand{\myplain}{%
\fancypagestyle{plain}{%
\fancyhead[LO]{\bfseries\rightmark}
\fancyhead[RE]{\bfseries\leftmark}
\renewcommand{\headrulewidth}{0.5pt}
}}
\plainplain
\pagestyle{plain}
\usepackage{lipsum, filecontents}
\begin{document}
\frontmatter
\pagenumbering{Roman}
\tableofcontents
\clearpage
\listoffigures
\clearpage
\listoftables
\clearpage
\mainmatter % implies arabic pagenumbering
\myplain
\pagestyle{plain}
\begin{spacing}{1.5}
\chapter{Introduction}
\lipsum[1-7]
\section{New Section}
\lipsum[8-14]
\clearpage
\plainplain
\pagestyle{plain}
\backmatter
\pagenumbering{roman}
\setcounter{page}{5}
\nocite{mybook1}
\nocite{mybook2}
\nocite{mybook3}
\nocite{mybook5}
\nocite{mybook6}
\nocite{mybook7}
\nocite{mybook8}
\nocite{mybook10}
\bibliographystyle{plainnat}
\begin{filecontents}{bibliography.bib}
\documentclass[12pt, a4paper, oneside, onecolumn]{book}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[ngerman]{babel}
\usepackage{setspace}
\usepackage[round]{natbib}
\usepackage{fancyhdr}
\pagestyle{fancy}
\renewcommand{\chaptermark}[1]{%
\markboth{#1}{}}
\renewcommand{\sectionmark}[1]{%
\markright{\thesection\ #1}}
\fancyhf{} % delete current header and footer
\fancyhead[LE,RO]{\bfseries\thepage}
\fancyhead[LO]{\bfseries\rightmark}
\fancyhead[RE]{\bfseries\leftmark}
\renewcommand{\headrulewidth}{0.5pt}
\renewcommand{\footrulewidth}{0pt}
\addtolength{\headheight}{0.5pt} % space for the rule
\newcommand{\plainplain}{%
\fancypagestyle{plain}{%
\renewcommand{\headrulewidth}{0pt}
\fancyhead[LO]{}
\fancyhead[RE]{}
}}
\newcommand{\myplain}{%
\fancypagestyle{plain}{%
\fancyhead[LO]{\bfseries\rightmark}
\fancyhead[RE]{\bfseries\leftmark}
\renewcommand{\headrulewidth}{0.5pt}
}}
\plainplain
\pagestyle{plain}
\usepackage{lipsum, filecontents}
\begin{document}
\frontmatter
\pagenumbering{Roman}
\tableofcontents
\clearpage
\listoffigures
\clearpage
\listoftables
\clearpage
\mainmatter % implies arabic pagenumbering
\myplain
\pagestyle{plain}
\begin{spacing}{1.5}
\chapter{Introduction}
\lipsum[1-7]
\section{New Section}
\lipsum[8-14]
\clearpage
\plainplain
\pagestyle{plain}
\backmatter
\pagenumbering{roman}
\setcounter{page}{5}
\nocite{mybook1}
\nocite{mybook2}
\nocite{mybook3}
\nocite{mybook5}
\nocite{mybook6}
\nocite{mybook7}
\nocite{mybook8}
\nocite{mybook10}
\bibliographystyle{plainnat}
\begin{filecontents}{bibliography.bib}
@book{mybook1,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
@book{mybook2,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
@book{mybook3,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
@book{mybook4,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
@book{mybook5,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
@book{mybook6,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
@book{mybook7,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
@book{mybook8,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
@book{mybook9,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
@book{mybook10,
title = {\lipsum[1]},
author = {myauthor},
year = {2010}
}
\end{filecontents}
\bibliography{bibliography}
\end{spacing}
\end{document}
Look at the incorrect result. It should not look like that. Also, the bibliography's last page is the only page that is wrong. All the other pages are like in my example of the correct version.

Question: Why does spacing change the header of my bibliography's last page?
lipsumorblindtext– Johannes_B Feb 04 '14 at 18:05lipsumfor text, but I didn't know that I could use the command in my bibliography as well. Makes sense, though, as it is part of thetexfile. I changed it, and now the code is shorter. – Xiphias Feb 04 '14 at 18:11\end{spacing}– Johannes_B Feb 04 '14 at 18:14TeX, but for this I do not have an explanation. – Xiphias Feb 04 '14 at 18:17spacingenvironment to begin with. – egreg Feb 04 '14 at 18:23TeX-like alternative here using\linespreaddirectly: http://tex.stackexchange.com/questions/61155/linespacing-without-packages – Xiphias Feb 04 '14 at 18:25spacingenvironment, not the package. – egreg Feb 04 '14 at 18:44spacingenvironment). Within that group you made changes to the headers. These changes are lost at the end of that group (at\end{spacing}). The last page is output after that group ends and so it gets the headers that were in effect before the group was started. Most things in (La)TeX are like this: changes made inside a group revert to their previous definitions at the end of the group. – Dan Feb 04 '14 at 21:06