TikZ to non-PDF

You have empty lines between the document and tikzpicture environment which puts the picture in an paragraph (which is \textwidth wide). Simply removing the lines fixes this:

\documentclass{standalone}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
  \draw [step=0.5] (-1.4,-1.4) grid (1.4,1.4);
\end{tikzpicture}
\end{document}

The rest of your question is already answered in Compile a LaTeX document into a PNG image that's as short as possible.

In short:

pdflatex file
convert -density 300 file.pdf -quality 90 file.png

or with v1.0 of the standalone class:

\documentclass[convert={density=300,size=1080x800,outext=.png}]{standalone}

compile with:

pdflatex -shell-escape file

You might also want to use the new border option to set the border to 0pt.

This is a complement to the Martin Scharrer's answer for the particular case of Windows. This answer is not exhaustif, but gives a particular configuration that works for me.

Installation under Windows (checked with XP/7/10)

1. Install GhostScript;
2. Put the folder (something like C:\...\Ghostscript\bin) containing gswin32c.exe in the PATH;
3. Create a folder named StackExchange where you will put all your questions and answers for TeX.SX;

4. In this folder create a file standalone.cfg with the following content

   \input{standalone/standalone.cfg}
   \standaloneconfig{convert={ghostscript, density=700, command={\gsexe\space -dDownScaleFactor=4 -dSAFER -dBATCH -dNOPAUSE -sDEVICE=\gsdevice\space -r\density\space -sOutputFile=\outfile\space \infile}}}
   

5. Use as model for your question/answer in this folder something like

   \documentclass[tikz,border=7pt]{standalone}
   % \usetikzlibrary{ }
   \begin{document}
     \begin{tikzpicture}
       \fill[red] (0,0) circle (1);
     \end{tikzpicture}
   \end{document}
   

And now just compile your tex file with the option -shell-escape and you will obtain a PNG with the same name and "resonable" default size.