Do not use package titlesec together with a KOMA-Script class. It is possible to get the same format of the chapter heading using KOMA-Script commands.
\renewcommand\chapterformat{%
\thechapter\hspace*{20pt}\textcolor{gray75}{|}\hspace*{20pt}%
}
\addtokomafont{chapter}{\Huge\rmfamily}
Additional remarks:
- Use the new syntax for option
bibliography=totoc.
- Remove option
pdftex for package graphicx.
- If there should be a
\parskip instead \parindent to mark new paragraphs use the class option parskip with one of the values full, half etc. (see the KOMA-Script documentation).
- There is no need to load package
caption in your example. You can change the font of the captions and their labels using \addtokomafont or \setkomafont.
hyperref should be the last package.

Code:
\documentclass[12pt,DIV = 15,titlepage,
bibliography=totoc % <- changed syntax
]{scrreprt}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage{fourier,avant}
\usepackage{graphicx} % <-remove option pdftex
\usepackage[onehalfspacing]{setspace}
\setlength{\parindent}{0pt} % <- do really want no paragraph mark?
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{textcomp}
\usepackage{wallpaper}
\usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry}
\usepackage{color}
\definecolor{gray75}{gray}{0.75}
\renewcommand\chapterformat{%
\thechapter\hspace*{20pt}\textcolor{gray75}{|}\hspace*{20pt}%
}
\addtokomafont{chapter}{\Huge\rmfamily}
\addtokomafont{caption}{\footnotesize}% <- instead using package caption
\addtokomafont{captionlabel}{\bfseries\sffamily}% <- <- instead using package caption
\usepackage{fancyhdr}
\pagestyle{fancy}
\renewcommand{\chaptermark}[1]{\markboth{#1}{#1}}
\renewcommand{\sectionmark}[1]{\markright{#1}}
\fancyhf{}
\fancyhead[L]{\leftmark}
\fancyhead[R]{\rightmark}
\cfoot{\thepage}
\usepackage{blindtext}% dummy text
\usepackage{hyperref}% last package
\begin{document}
\chapter{works}
\subsection*{Doesn't}
Why not?
\begin{figure}[htb]
\centering
\rule{3cm}{2cm}%
\caption{A caption}%
\end{figure}
\blinddocument
\end{document}
Note that if you are using fancyhdr with a KOMA-Script class, you have to redefine \fancyhead and \fancyfoot (like in your code) to remove the old font commands like \sl etc.
But the recommended package for headers and footers in a KOMA-Script document is scrlayer-scrpage.
\documentclass[12pt,DIV = 15,titlepage,
bibliography=totoc % <- changed syntax
]{scrreprt}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage{fourier,avant}
\usepackage{graphicx} % <-remove option pdftex
\usepackage[onehalfspacing]{setspace}
\setlength{\parindent}{0pt} % <- do really want no paragraph mark?
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{textcomp}
\usepackage{wallpaper}
\usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry}
\usepackage{color}
\definecolor{gray75}{gray}{0.75}
\renewcommand\chapterformat{%
\thechapter\hspace*{20pt}\textcolor{gray75}{|}\hspace*{20pt}%
}
\addtokomafont{chapter}{\Huge\rmfamily}
\addtokomafont{caption}{\footnotesize}% <- instead using package caption
\addtokomafont{captionlabel}{\bfseries\sffamily}% <- <- instead using package caption
\usepackage[autooneside=false,automark,headsepline]{scrlayer-scrpage}
\clearpairofpagestyles
\ihead{\leftmark}
\ohead{\rightmark}
\cfoot*{\pagemark}
\addtokomafont{pageheadfoot}{\upshape}
\renewcommand\chaptermarkformat{}
\renewcommand\sectionmarkformat{}
\usepackage{blindtext}% dummy text
\usepackage{hyperref}
\begin{document}
\chapter{works}
\subsection*{Doesn't}
Why not?
\begin{figure}[htb]
\centering
\rule{3cm}{2cm}%
\caption{A caption}%
\end{figure}
\blinddocument
\end{document}
Maybe you can remove package wallpaper too because you can define an additional layer to use a background image.
If you really want to use titlesec together with a KOMA-Script class and you have the titlesec version 2.10.2 then you must install the prerelease of KOMA-Script 3.20. If your titlesec is version 2.10.1 you have to update this package too.
titlesecandfancyhdrand KOMA --- oh my ;-) – May 01 '16 at 18:23titlesec(and\titleformat) will cure the problem, but don't usefancyhdrwith KOMA or drop KOMA.hyperrefshould be the last package to be loaded here, not somewhere in between – May 01 '16 at 18:27