Issue inserting image in header

Are you sure that the images are not being pushed off the top off the page? Without your full directory, I am not able to reproduce your problem but it looks like you haven't specified a large enough top value within the geometry argument. Looking at the documentation to the geometry LaTeX function, you can see that this specifies the distance between the top of the body and the edge of the paper, leaving no space for your header.

Here is my reproducible example. It generates a dummy plot plot.png which is used for a logo placeholder

index.Rmd:

---
documentclass: book
classoption: openany
fontsize: 12pt
linestretch: 1.25
link-citations: yes
output:
  bookdown::pdf_book:
    includes:
      in_header: preamble.tex
    latex_engine: xelatex
    toc_depth: 4
subparagraph: yes
---

```{r setup, include=FALSE}
# Create an image to attach
png(filename = "logo.png", width = 200, height = 150, units = "px")
plot(cars)
box('figure', col = 'red')
dev.off()
```

# Title 1

Text

\newpage

## Section

Text

\newpage

Within the same directory is the preamble.tex. Note the adjusted geometry argument top=50mm:

\usepackage[utf8]{inputenc}
\usepackage{graphicx}
\usepackage{geometry}
\usepackage{titlesec}
\usepackage{fancyhdr}
\usepackage{fontspec}
\setmainfont{Arial}

\pagestyle{fancy}\setlength\headheight{100pt}
\fancyhead[L]{\includegraphics[width=4cm]{logo.png}}
\fancyhead[R]{\includegraphics[width=4.1cm]{logo.png}}
\renewcommand{\headrulewidth}{0pt}
\rfoot{Seite \thepage \hspace{1pt} von \pageref{LastPage}}

\geometry{a4paper, total={170mm,257mm}, left=20mm, top=50mm}

\titleformat{\chapter}
  {\normalfont\LARGE\bfseries}{\thechapter}{1em}{}
\titlespacing*{\chapter}{0pt}{3.5ex plus 1ex minus .2ex}{2.3ex plus .2ex}

Edit: I seem to have had some kind of bug in the system despite reloading everything. I followed the suggestions by Michael Harper but opened the code in a totally new project/folder, then I set the documentclass: article inserted a title page manually. What is left now: Getting the headers to work in the tableofcontents and removing the page number (footer, centered).

Index.Rmd:

---
documentclass: article
classoption: openany
site: bookdown::bookdown_site
fontsize: 12pt
linestretch: 1.25
link-citations: yes
output:
  bookdown::pdf_book:
    includes:
      in_header: preamble.tex
      before_body: before_body.tex
    latex_engine: xelatex
    toc: yes
    toc_depth: 4
subparagraph: yes
---

```{r setup, include=FALSE}
# Create an image to attach
png(filename = "logo.png", width = 200, height = 150, units = "px")
plot(cars)
box('figure', col = 'red')
dev.off()
```

preamble.tex:

% loading necessary LateX-packages
\usepackage[utf8]{inputenc}
\usepackage{graphicx}
\usepackage{geometry}
\usepackage{titlesec}
\usepackage{fancyhdr}
\usepackage{fontspec}
\setmainfont{Arial}

% include a fancy header
\pagestyle{fancy}\setlength\headheight{100pt}
\fancyhead[L]{\includegraphics[width=5cm]{logo.png}}
\fancyhead[R]{\includegraphics[width=4.1cm]{logo.png}}
\renewcommand{\headrulewidth}{0pt}
\rfoot{Seite \thepage \hspace{1pt} von \pageref{LastPage}}

% set page geometry
\geometry{a4paper, total={170mm,257mm}, left=20mm, top=40mm}

% make the word "Chapter" disappear - only display the chapters name
\titleformat{\chapter}
{\normalfont\LARGE\bfseries}{\thechapter}{1em}{}
\titlespacing*{\chapter}{0pt}{3.5ex plus 1ex minus .2ex}{2.3ex plus .2ex}

% Following below
% turns maketitle off, then you can define your own titlepage in before_body.tex
\let\oldmaketitle\maketitle 
\AtBeginDocument{\let\maketitle\relax}

before_body.tex:

% titlepage
\thispagestyle{empty}
\begin{center}
{\Huge A BOOK}
\linebreak
\includegraphics{cover.png}
\linebreak
{\huge by Me}
\end{center}

\let\maketitle\oldmaketitle
\maketitle