Reducing spacing after headings

You've loaded the titlesec package- it provides the command \titlespacing which has the format

\titlespacing{command}{left spacing}{before spacing}{after spacing}[right]

From the titlesec package

% spacing: how to read {12pt plus 4pt minus 2pt}
%           12pt is what we would like the spacing to be
%           plus 4pt means that TeX can stretch it by at most 4pt
%           minus 2pt means that TeX can shrink it by at most 2pt
%       This is one example of the concept of, 'glue', in TeX

A complete MWE follows

\documentclass{article}
\usepackage{lipsum}
\usepackage{titlesec}

\titlespacing\section{0pt}{12pt plus 4pt minus 2pt}{0pt plus 2pt minus 2pt}
\titlespacing\subsection{0pt}{12pt plus 4pt minus 2pt}{0pt plus 2pt minus 2pt}
\titlespacing\subsubsection{0pt}{12pt plus 4pt minus 2pt}{0pt plus 2pt minus 2pt}

\begin{document}
\section{Section}
\lipsum[1]

\end{document}

You can use the \titlespacing command from the titlesec package. For example:

\titlespacing{\section}{0pt}{\parskip}{-\parskip}
\titlespacing{\subsection}{0pt}{\parskip}{-\parskip}
\titlespacing{\subsubsection}{0pt}{\parskip}{-\parskip}

See titlesec.pdf for more options.

If you are using setspace, then mixing it with your own \linespread{1.5} is probably not a good idea.

Try to delete the line about \linespread and use instead

\usepackage{setspace}
\onehalfspacing

The package documentation says that it does more than just change line spread: it also take care of skips before and after headings, displays, etc. - the things you say you need.