I think you want something like this, the key is to use \markright (or \markboth or one of the other mark commands) to leave a mark on the page, the page heading can then pick up the first and last mark on the page and use accordingly. In the standard classes \section uses \markright (with chapter level marks using the left mark) so I did the same here, defining a simple \section that uses a counter and leaves a mark but no optional handling or star form, and very little heading formatting

\documentclass{report}
\usepackage[paperheight=8cm,paperwidth=7cm, margin=2cm]{geometry}
\pagestyle{headings}
\renewcommand\section[1]{%
\refstepcounter{section}%
\textsection\thesection\markright{#1}~\textit{#1} %
}
\begin{document}
%\chapter{zzz} omit this as page heading is open on chapter start pages by default
\refstepcounter{chapter}
\section{Start} something about the beginning \section{This} and a bit more
\section{That} A longer section about something that has more than one paragraph.
It seems a bit odd to have a paragraph more distinguished than a section heading, so perhaps that is not needed. \section{Then} and sections will always be inline.
\section{Later} or at the start of a paragraph.
\section{Second Start} something about the beginning \section{Second This} and a bit more
\section{Second That} A longer section about something that has more than one paragraph.
It seems a bit odd to have a paragraph more distinguished than a section heading, so perhaps that is not needed. \section{Second Then} and sections will always be inline.
\section{Second Later} or at the start of a paragraph.
\section{Third} Is surprisingly different to the first two.
\end{document}