How to do fancy color background header and footer in a book?

Try this:

documentclass[11pt,a4paper,openright,fleqn,twoside]{book}

usepackage[paperwidth=16cm,paperheight=24cm]{geometry}
usepackage{titleps}
usepackage{lipsum}

newpagestyle{mystyle}{%
    headrule%
    sethead[includeHeaderImagebfseriesthepage][][bfseriesthechapter.~chaptertitle]
      {includeHeaderImagebfseriesthechapter. ~chaptertitle}
      {}{bfseriesthepage}
    setfoot[includeFooterImage][][]{includeFooterImage}{}{}
}
pagestyle{mystyle}

usepackage{tikz}

makeatletter
newcommandincludeHeaderImage{%
  includeImageOnPage
    {@headerimage}{width=paperwidth, height=50pt}{north east}%
}

newcommandincludeFooterImage{%
  includeImageOnPage
    {@footerimage}{width=paperwidth, height=50pt}{south east}%
}

% #1 = a cmd containing figure file, #2 = figure options, #3 = tikz node anchor
newcommandincludeImageOnPage[3]{%
  ifx#1@empty
  else
    edef@tempa{%
      noexpandtikz[overlay, remember picture]
        noexpandnode[inner sep=0pt, anchor=#3] at (current page.#3)
          {noexpandincludegraphics[#2]{#1}};%
    }
    @tempa
  fi
}

newcommandsetHeaderImage[1]{def@headerimage{#1}}
newcommandsetFooterImage[1]{def@footerimage{#1}}

% init
setHeaderImage{}
setFooterImage{}
makeatother
begin{document}
    
chapter{First Chapter}
lipsum[1]
newpage

% set new header and footer images
setHeaderImage{example-image-a}
setFooterImage{example-image-b}
section{First Section}
lipsum[3-4]
newpage

% cancel
setHeaderImage{}
setFooterImage{}
subsection{First Subsection}
lipsum[5-6]
end{document}