I would like to reduce some vertical space near the section title (above and
below the section title). Could you share some hints?

I put the following in the LaTeX preamble section of the Document Settings. This allows for a one-spot change that makes changes to the whole document. This is really more of a LaTeX solution than a LyX solution, but works well. You can modify the text styles and spacing as needed. Note that I use this with the article class.

\renewcommand{\section}{ %
\...@startsection{section}{1}{0pt} %
{1.1ex plus 2.0ex minus 1.0ex}%
{1.2ex plus 2.0ex minus 1.0ex} %
{\bfseries \large} %
}
\renewcommand{\subsection}{ %
\...@startsection{subsection}{2}{0pt} %
{1.1ex plus 2.0ex minus 1ex}%
{1.2ex plus 2.0ex minus 1ex} %
{\bfseries} %
}
\renewcommand{\subsubsection}{ %
\...@startsection{subsubsection}{3}{0pt} %
{1.1ex plus 2.0ex minus 0.5ex}%
{1.2ex plus 2.0ex minus 0.5ex} %
{\slshape} %
}
\renewcommand{\paragraph}{ %
\...@startsection{paragraph}{4}{0pt} %
{1.25ex plus 1.0ex minus 0.5ex}%
{-0.5em} %
{\bfseries} %
}

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