Setting a fixed space between header text and head rule AND also between footer text and foot rule

Question

Problem Description

I wish to set a fixed space between header/footer text and the headrule/footrule. I have gone through Setting same distance between header rule, footer rule and text and it is not what I want exactly. For the headers, the space between the last line of the long header text and the headrule is different from the space between single lined header text and the corresponding headrule. In other words, can we fix this spacing so that it remain the same for both single line (short header text) and multiline (long header text) header text? Given that the solution to the header text and headrule has been found, I want to apply that fixed spacing used between headrule and header text to footrule and footer text as well. Below is my MWE:

% !TeX TS-program = pdflatex    
\documentclass[12pt,openany,twoside]{book}
\raggedbottom
\usepackage[ 
left=2.5cm,
right=2.5cm, 
top=3cm, 
bottom=3cm,
headheight = 3.5\baselineskip,
headsep = 5mm,
a4paper
]{geometry}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage[table]{xcolor}
\usepackage{graphicx}
\newcommand\HRule{\noindent\rule{\linewidth}{1.5pt}}
\usepackage{background}
\backgroundsetup{contents={}}
\usepackage{fancyhdr}
\usepackage{hhline} 

\usepackage{helvet}


\usepackage{tabularx}%      
\newcommand*\myheaderfooterfont{\sffamily\bfseries\selectfont}  

\newlength{\leftspace}% for sections <<<<<<<<<
\setlength{\leftspace}{20ex}%set

\newlength{\Lheader}
\renewcommand{\chaptermark}[1]{\markboth{\uppercase{#1}}{}}%  
\renewcommand{\sectionmark}[1]{\markright{\uppercase{#1}}}%

\fancypagestyle{fancy}{%
    \fancyhf{}  
    \fancyhead[OL]{\myheaderfooterfont%
        \ifnum\value{chapter}>0%
        \begin{tabularx}{\textwidth}{@{}lX}\uppercase{Chapter~\thechapter}&\raggedright\leftmark\end{tabularx}% changed <<<<<<<<<<
        \else\raggedright\leftmark\relax\fi}
    \fancyhead[ER]{\myheaderfooterfont%
        \settowidth{\Lheader}{\thesection\hspace*{12pt}\rightmark}  
        \ifnum\value{chapter}>0%
        \ifdim\Lheader>\linewidth%          
        \begin{tabularx}{\dimexpr\textwidth-\leftspace}{@{}lX} \thesection &\raggedright\rightmark\end{tabularx}%
        \else\begin{tabular}{@{}ll@{}} \thesection &\raggedright\rightmark\end{tabular}%
        \fi\else\hfill\raggedright\rightmark\relax\fi
    }% changed <<<<<
    
    \fancyfoot{}
    \fancyfoot[RE,RO]{\myheaderfooterfont\thepage}%last added<<<<<<<
%<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<last added
    \renewcommand\headrule
{{     \color{blue}\rule{0.33\headwidth}{2.5pt}%   <---
       \color{red}\rule{0.34\headwidth}{2.5pt}%    <---
       \color{orange}\rule{0.33\headwidth}{2.5pt}% <---
}}
%<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<last added
}

\fancypagestyle{plain}{%
    \fancyhf{}%
    \renewcommand\headrule
    {{     \color{blue}\rule{1\headwidth}{0pt}%         
    }}   
    \fancyfoot{}
    \fancyfoot[RE,RO]{\myheaderfooterfont\thepage} %last added<<<<<<  
}

\pagestyle{fancy}   
\renewcommand{\footrulewidth}{1pt}

\usepackage{showframe}

\usepackage[none]{hyphenat}
\raggedright %  important with  none hyphenat + none !!!!!!!!!!!!!!!!!

\begin{document} 
%<<<<<<<<<<<<<<<<<<<<<<<<<last added
    \begin{titlepage}
        \setlength{\parindent}{0in} 
        \topskip 360pt
        \vspace*{\fill}
        
        \centering 
        
        \scshape 
     
        
        
        \begin{flushleft}
            
            { \textbf{HHHHHHHHHHHHHHHHHHHHHHHHGFFFFFFFFFFF}
            }
            
        \end{flushleft}
        
        \vspace*{3\baselineskip} 
        
        
        
        \textbf{AUTHOR}
        
        
        \vspace{0.5\baselineskip} 
        
        {\scshape\Large\selectfont LKJJHHHG} 
        
        \vspace{1.5\baselineskip} 
        
        
        
        \textbf{PFEYTYU}
        
        
        \vspace{1\baselineskip} 
        
        {\scshape\Large\selectfont JKIU} 
        
        \vspace{2\baselineskip} 
        
        
        
        \textbf{YUUUUH}
        
        
        
        
        \vspace{0.5\baselineskip} 
        
        {\scshape\Large\selectfont 12345} 
        
        \vspace{0.5\baselineskip} 
        
        \vfill 
        
        
        
        
        \vspace{4\baselineskip} 
        \scshape 
        Revised: \today 
        
        
    \end{titlepage}
    %<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<last added
    \frontmatter
    \chapter*{ABC}
    
    \markboth{ABC1}{ABC2}
    
    eeeheeyeyeyeyeyeeyeye
    \section*{DEG}
    dddheheheheheheheehehehe
    \pagebreak
    \section*{KJ}
    ddhheheheheheehehehheheheheheh
    \pagebreak
    \section*{KDH}
    dhddhhehehehehehehehehehehe
    
    \mainmatter
    
    \chapter{ABCD ffhfgrhr ejejejeje fhfhfhf ehehehehe dhddhdhellgl rkttkgkhihi eeeejejwnwn hhlhlnnmgjgj rjrrjoaoaoao  }        
    eehehdhdhdhdddddd ddddddddddddddddddddd     
    
    \newpage                
    \section{Special keyboard characters gghthr rhrrhd fheheh fhehuw fhewiwi feiwiwd fiiwiwid fiiwiwi}  
    
    \LaTeX{} hasmany symbols at its disposal. The majority of them are within the mathematical domain as shown in \eqref{eq:1} and \eqref{eq:2}. The ten special keyboard characters are: \verb|#|, \verb|$|, \verb|%|, \verb|&|, \verb|~|, \verb|_|, \verb|^|, \verb|\| and  \verb|{}|. If you simply want the character to be printed just as any other letter, include a \verb|\ | in front of the character. For example, \verb|\$| will produce \verb|$| in your output.
    The exception to the rule is the \verb|\ | itself because \verb|\\ | has its own special meaning. A \verb|\ | is produced by typing \verb|$\backslash$| in your file.       
    \newpage        
    \section{Special keyboard characters}       
    Special keyboard characters dhgdgdg dgdgdgdg ggdgdgd egegegege dgddgdgd eggegegeg dgdgdgdgd egegegeg dggeieieirr tktii43iwje rfrhrhrhr gtktotyiyighgjgj fkf     
    \newpage        
    Special keyboard characters dhgdgdg dgdgdgdg ggdgdgd egegegege dgddgdgd eggegegeg dgdgdgdgd egegegeg dggeieieirr tktii43iwje rfrhrhrhr gtktotyiyighgjgj fkf
    
    \chapter{One short line}        
    \LaTeX{} has many symbols at its disposal. The majority of them are within the mathematical domain as shown in \eqref{eq:1} and \eqref{eq:2}. The ten special keyboard characters are: \verb|#|, \verb|$|, \verb|%|, \verb|&|, \verb|~|, \verb|_|, \verb|^|, \verb|\| and  \verb|{}|. If you simply want the character to be printed just as any other letter, include a \verb|\ | in front of the character. For example, \verb|\$| will produce \verb|$| in your output.
    The exception to the rule is the \verb|\ | itself because \verb|\\ | has its own special meaning. A \verb|\ | is produced by typing \verb|$\backslash$| in your file. 
    
    \section{Special keyboard characters}   
    Special keyboard characters dhgdgdg dgdgdgdg ggdgdgd egegegege dgddgdgd eggegegeg dgdgdgdgd egegegeg dggeieieirr tktii43iwje rfrhrhrhr gtktotyiyighgjgj fkf     
    \newpage        
    Special keyboard characters dhgdgdg dgdgdgdg ggdgdgd egegegege dgddgdgd eggegegeg dgdgdgdgd egegegeg dggeieieirr tktii43iwje rfrhrhrhr gtktotyiyighgjgj fkf     
    \newpage        
    Special keyboard characters dhgdgdg dgdgdgdg ggdgdgd egegegege dgddgdgd eggegegeg dgdgdgdgd egegegeg dggeieieirr tktii43iwje rfrhrhrhr gtktotyiyighgjgj fkf     
    
\end{document}

Answer 1

EDIT: Code adapted to the latest MWE. Complete code added. See below.

I'll explain what is happening. Your \headrule uses the \rule command. This is a "horizontal" command, which means it becomes part of the paragraph that contains the header text. It goes to a new line of this paragraph because it wouldn't fit next to the header text, and of course, that is what you want. But TeX then makes sure that the distance of the baseline of the header text and the baseline of this rule is at least \baselineskip. That is why there is such a big distance between the header and the rule in the single line header.

In the case of the multiline header with \tabular(x), the baseline of the header is the first line of the multiline header. So the distance of the header baseline and the rule is already greater than \baselineskip and no extra space is added.

The standard definition of \headrule in fancyhdr use the \hrule command, which is a vertical mode command. Therefore it is not part of the paragraph and it is just tacked to the bottom of the header text. (The distance between the baseline and the rule is guaranteed by a \strut.)

So one solution would be to define your \headrule with \hrule:

\renewcommand\headrule{\color{blue}\hrule height 2.5pt}

Another possibility is to include the command \nointerlineskip before your \rule. This tells Tex not to add the extra \baselineskip.

\renewcommand\headrule{\nointerlineskip\color{blue}\rule{\headwidth}{2.5pt}}

Now for the footer:

The distance between the \footrule and the footer text is defined by footruleskip. This is not the distance between the rule and the top of the text, however, because there is also a \strut in the footer. This id done to get a uniform distance between the footrule and the baseline of the footer. Otherwise this distance would depend on whether there are capital letters in the footer or only lowercase letters. I think this would not be aesthetic. So I suggest to keep this. It would be difficult to remove the \strut anyway. In your case if you have only numbers in the footer (and in this font they all have the same height), it wouldn't matter.

You could measure the height of the digits, but I think just eyeballing is good enough. I suggest:

\renewcommand{\footruleskip}{3pt}

Complete code (my changes indicated by %%%%%%%% PVO %%%%%%%%%%)

\documentclass[12pt,openany,twoside]{book}
\raggedbottom
\usepackage[ 
left=2.5cm,
right=2.5cm, 
top=3cm, 
bottom=3cm,
headheight = 3.5\baselineskip,
headsep = 5mm,
a4paper
]{geometry}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage[table]{xcolor}
\usepackage{graphicx}
\newcommand\HRule{\noindent\rule{\linewidth}{1.5pt}}
\usepackage{background}
\backgroundsetup{contents={}}
\usepackage{fancyhdr}
\usepackage{hhline} 

\usepackage{helvet}


\usepackage{tabularx}%      
\newcommand*\myheaderfooterfont{\sffamily\bfseries\selectfont}  

\newlength{\leftspace}% for sections <<<<<<<<<
\setlength{\leftspace}{20ex}%set

\newlength{\Lheader}
\renewcommand{\chaptermark}[1]{\markboth{\uppercase{#1}}{}}%  
\renewcommand{\sectionmark}[1]{\markright{\uppercase{#1}}}%

\fancypagestyle{fancy}{%
    \fancyhf{}  
    \fancyhead[OL]{\myheaderfooterfont%
        \ifnum\value{chapter}>0%
        \begin{tabularx}{\textwidth}{@{}lX}\uppercase{Chapter~\thechapter}&\raggedright\leftmark\end{tabularx}% changed <<<<<<<<<<
        \else\raggedright\leftmark\relax\fi}
    \fancyhead[ER]{\myheaderfooterfont%
        \settowidth{\Lheader}{\thesection\hspace*{12pt}\rightmark}  
        \ifnum\value{chapter}>0%
        \ifdim\Lheader>\linewidth%          
        \begin{tabularx}{\dimexpr\textwidth-\leftspace}{@{}lX} \thesection &\raggedright\rightmark\end{tabularx}%
        \else\begin{tabular}{@{}ll@{}} \thesection &\raggedright\rightmark\end{tabular}%
        \fi\else\hfill\raggedright\rightmark\relax\fi
    }% changed <<<<<
    
    \fancyfoot{}
    \fancyfoot[RE,RO]{\myheaderfooterfont\thepage}%last added<<<<<<<
%<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<last added
    \renewcommand\headrule
{{     \nointerlineskip %%%%%%%% PVO %%%%%%%%%%
       \color{blue}\rule{0.33\headwidth}{2.5pt}%   <---
       \color{red}\rule{0.34\headwidth}{2.5pt}%    <---
       \color{orange}\rule{0.33\headwidth}{2.5pt}% <---
}}
%<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<last added
}

\fancypagestyle{plain}{%
    \fancyhf{}%
    \renewcommand\headrule
    {{     \color{blue}\rule{1\headwidth}{0pt}%         
    }}   
    \fancyfoot{}
    \fancyfoot[RE,RO]{\myheaderfooterfont\thepage} %last added<<<<<<  
}

\pagestyle{fancy}   
\renewcommand{\footrulewidth}{1pt}
\renewcommand{\footruleskip}{3pt} %%%%%%%% PVO %%%%%%%%%%

\usepackage{showframe}

\usepackage[none]{hyphenat}
\raggedright %  important with  none hyphenat + none !!!!!!!!!!!!!!!!!

\begin{document} 
%<<<<<<<<<<<<<<<<<<<<<<<<<last added
    \begin{titlepage}
        \setlength{\parindent}{0in} 
        \topskip 360pt
        \vspace*{\fill}
        
        \centering 
        
        \scshape 
     
        
        
        \begin{flushleft}
            
            { \textbf{HHHHHHHHHHHHHHHHHHHHHHHHGFFFFFFFFFFF}
            }
            
        \end{flushleft}
        
        \vspace*{3\baselineskip} 
        
        
        
        \textbf{AUTHOR}
        
        
        \vspace{0.5\baselineskip} 
        
        {\scshape\Large\selectfont LKJJHHHG} 
        
        \vspace{1.5\baselineskip} 
        
        
        
        \textbf{PFEYTYU}
        
        
        \vspace{1\baselineskip} 
        
        {\scshape\Large\selectfont JKIU} 
        
        \vspace{2\baselineskip} 
        
        
        
        \textbf{YUUUUH}
        
        
        
        
        \vspace{0.5\baselineskip} 
        
        {\scshape\Large\selectfont 12345} 
        
        \vspace{0.5\baselineskip} 
        
        \vfill 
        
        
        
        
        \vspace{4\baselineskip} 
        \scshape 
        Revised: \today 
        
        
    \end{titlepage}
    %<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<last added
    \frontmatter
    \chapter*{ABC}
    
    \markboth{ABC1}{ABC2}
    
    eeeheeyeyeyeyeyeeyeye
    \section*{DEG}
    dddheheheheheheheehehehe
    \pagebreak
    \section*{KJ}
    ddhheheheheheehehehheheheheheh
    \pagebreak
    \section*{KDH}
    dhddhhehehehehehehehehehehe
    
    \mainmatter
    
    \chapter{ABCD ffhfgrhr ejejejeje fhfhfhf ehehehehe dhddhdhellgl rkttkgkhihi eeeejejwnwn hhlhlnnmgjgj rjrrjoaoaoao  }        
    eehehdhdhdhdddddd ddddddddddddddddddddd     
    
    \newpage                
    \section{Special keyboard characters gghthr rhrrhd fheheh fhehuw fhewiwi feiwiwd fiiwiwid fiiwiwi}  
    
    \LaTeX{} hasmany symbols at its disposal. The majority of them are within the mathematical domain as shown in \eqref{eq:1} and \eqref{eq:2}. The ten special keyboard characters are: \verb|#|, \verb|$|, \verb|%|, \verb|&|, \verb|~|, \verb|_|, \verb|^|, \verb|\| and  \verb|{}|. If you simply want the character to be printed just as any other letter, include a \verb|\ | in front of the character. For example, \verb|\$| will produce \verb|$| in your output.
    The exception to the rule is the \verb|\ | itself because \verb|\\ | has its own special meaning. A \verb|\ | is produced by typing \verb|$\backslash$| in your file.       
    \newpage        
    \section{Special keyboard characters}       
    Special keyboard characters dhgdgdg dgdgdgdg ggdgdgd egegegege dgddgdgd eggegegeg dgdgdgdgd egegegeg dggeieieirr tktii43iwje rfrhrhrhr gtktotyiyighgjgj fkf     
    \newpage        
    Special keyboard characters dhgdgdg dgdgdgdg ggdgdgd egegegege dgddgdgd eggegegeg dgdgdgdgd egegegeg dggeieieirr tktii43iwje rfrhrhrhr gtktotyiyighgjgj fkf
    
    \chapter{One short line}        
    \LaTeX{} has many symbols at its disposal. The majority of them are within the mathematical domain as shown in \eqref{eq:1} and \eqref{eq:2}. The ten special keyboard characters are: \verb|#|, \verb|$|, \verb|%|, \verb|&|, \verb|~|, \verb|_|, \verb|^|, \verb|\| and  \verb|{}|. If you simply want the character to be printed just as any other letter, include a \verb|\ | in front of the character. For example, \verb|\$| will produce \verb|$| in your output.
    The exception to the rule is the \verb|\ | itself because \verb|\\ | has its own special meaning. A \verb|\ | is produced by typing \verb|$\backslash$| in your file. 
    
    \section{Special keyboard characters}   
    Special keyboard characters dhgdgdg dgdgdgdg ggdgdgd egegegege dgddgdgd eggegegeg dgdgdgdgd egegegeg dggeieieirr tktii43iwje rfrhrhrhr gtktotyiyighgjgj fkf     
    \newpage        
    Special keyboard characters dhgdgdg dgdgdgdg ggdgdgd egegegege dgddgdgd eggegegeg dgdgdgdgd egegegeg dggeieieirr tktii43iwje rfrhrhrhr gtktotyiyighgjgj fkf     
    \newpage        
    Special keyboard characters dhgdgdg dgdgdgdg ggdgdgd egegegege dgddgdgd eggegegeg dgdgdgdgd egegegeg dggeieieirr tktii43iwje rfrhrhrhr gtktotyiyighgjgj fkf     
    
\end{document}