Customize KOMA section headings without titlesec

Question

A quick question: Is it possible to change the name of a part/chapter/section with some sort of command?

For example, I have a chapter called "Important Things". I created this chapter with the command \chapter{Important Things}. This prints the title of the chapter and adds an entry in the ToC.

I guess that the information about the chapter name is stored and can somehow be accessed by a command like \chaptername or \chaptertitle or something similar. So let's assume that "Important Things" is the current "value" of the chapter-name. What if I now want to change this "value" into "Unimportant Things". For this I cannot use \chapter{Unimportant Things}, because if I use that, then I change the value of the chapter-name but I also print a chapter heading.

Maybe I could also ask the following: Can I create a part/chapter/section without printing a heading and without having an entry in the ToC?

I am not sure if I made myself clear.

I've been asked for a MWE. There you see that the page headers after the new part continue to bear the section title and chapter title of the previous part.

%!TEX TS-program = lualatexmk
\documentclass{scrbook}
\usepackage{fmtcount}
\usepackage[noindentafter]{titlesec}

    \addtokomafont{part}{\thispagestyle{empty}\Huge\textsc}
    \addtokomafont{partnumber}{\Huge\textsc}
    \renewcommand*{\partformat}{\partname~\numberstring{part}}

    \titleformat{\chapter}[display]{\fillast}{\Large\textsc{\chaptertitlename~\numberstring{chapter}}}{1ex minus .1ex}{\LARGE\emph}
    \titlespacing{\chapter}{1pc}{*3}{*20}[1pc]

    \titleformat{\section}[block]{\fillast\large}{\thesection}{.5em}{\large\textsc}
    \titlespacing{\section}{1pc}{*6}{*2}[1pc]

    \titleformat{\subsection}[block]{\fillast\itshape}{\thesubsection}{.5em}{}
    \titlespacing{\subsection}{1pc}{*3}{*2}[1pc]

    \titleformat{\subsubsection}[block]{\fillast}{}{0pt}{}
    \titlespacing{\subsubsection}{1pc}{*3}{*2}[1pc]

\usepackage{titleps}
    \newpagestyle{chapter_and_section}{
        \sethead[\thepage][\emph{\chaptertitle}][]{}{\emph{\sectiontitle}}{\thepage}
        \setfoot[][][]{}{}{}
            }
        \pagestyle{chapter_and_section}
    \newpagestyle{first_page_of_chapter}{   % here is the definition of the first page of a chapter
        \sethead[][][]{}{}{}
        \setfoot[][\thepage][]{}{\thepage}{}
            }
        \assignpagestyle{\chapter}{first_page_of_chapter}

\begin{document}
\addpart{Some Things}
\chapter{Some Stuff}
\section{More Stuff}
Text.\newpage
Text.\newpage
Text.\newpage
Text.\newpage
\addpart{Important Things}
Text.\newpage
Text.\newpage
Text.\newpage
Text.\newpage
\end{document}

Answer 1

Using the interface of scrlayer-scrpage provided by the KOMA-bundle, this should be what you are after:

\documentclass{scrbook}
\usepackage{scrlayer-scrpage}
\ohead{\pagemark}
\cehead{\leftmark}
\cohead{\rightmark}

\renewcommand{\chaptermarkformat}{}
\renewcommand{\sectionmarkformat}{}

\begin{document}
\addpart{Some Things}
\chapter{Some Stuff}
\section{More Stuff}
Text.\newpage
Text.\newpage
Text.\newpage
Text.\newpage
\addpart{Important
Things}
Text.\newpage
Text.\newpage
Text.\newpage
Text.\newpage
\end{document}

Only KOMA

Taking into account the other stuff with the ambition not to use packages like titlesec or titleps which are incompatible to KOMA-script, which has its own interface.

Here should be what you want using only KOMA-script stuff. I should mention that it uses features that have just been introduced with KOMA-script v. 3.15.

Below, i tried to put the stuff and thinking process into blocks. For example, first adjusting the space above chapters, and at last the space below chapters.

\documentclass{scrbook}
%\usepackage{fmtcount}%not needed
\usepackage{blindtext}

\addtokomafont{part}{\Huge\textsc}
\renewcommand{\partpagestyle}{empty}
\addtokomafont{partnumber}{\Huge\textsc}

%dealing with chapters
\KOMAoption{chapterprefix}{true}
\let\raggedchapter\centering%Center all chapterheads
\addtokomafont{disposition}{\normalfont}%removes the bold face
\renewcommand{\chapterheadstartvskip}{\vspace{3ex}}
\addtokomafont{chapterprefix}{\Large\scshape}
\renewcommand{\chapterheadmidvskip}{\par\nobreak\vspace{1ex minus .1ex}}
\setkomafont{chapter}{\LARGE\emph}
\renewcommand{\chapterheadendvskip}{\vspace{20ex}}

%dealing with (sub/subsub)sections
\let\raggedsection\centering%Center all sectioningheads
%all levels have something in common, let's save typing:
\RedeclareSectionCommands[beforeskip=-3ex,
afterskip=2ex]{section,subsection,subsubsection}
\addtokomafont{section}{\large\textsc}
\RedeclareSectionCommand[beforeskip=-6ex]{section}
\addtokomafont{subsection}{\itshape}

\usepackage[automark]{scrlayer-scrpage}
\ohead{\pagemark}
\cehead{\leftmark}
\cohead{\rightmark}
\renewcommand{\chaptermarkformat}{}
\renewcommand{\sectionmarkformat}{}
\ofoot*{}%Clear the outer foot
\cfoot[\pagemark]{}%pagemark in the center only on plain pages
\addtokomafont{pagehead}{\itshape}
%italics instead of slanted


\setkomafont{descriptionlabel}{\usekomafont{disposition}\scshape}
%descriptionlabels will now look a bit pleasing with respect to the headings

\begin{document}
\addpart{Some Things}
\blinddocument
\addpart{Important Things}
\blinddocument
\end{document}

Since you aren't used to the interface, it might be a bit intimidating. But i myself spent more time in the titlesec documentation to look up which space is where. You should have no trouble figuring out where to change which space/skip.

Addendum:

The pagehead was set to itshape but it looked almost identical before?
Everytime one can set or add to a komafont it will be preset somewhere. For the pagehead it is \normalfont\slshape.