If you are willing to use an additional package, I'd suggest you to use tcolorbox. This offers you endless customization possibilities and offers you mechanisms to solve the issues you mention (no page breaks in the title, minimum 5 lines before a page break, automatic handling of the optional argument, easily customization of spacing betwee the various elements).
Here's a little example emulating your environment:
\documentclass{article}\usepackage[a5paper]{geometry}% just for the example\usepackage{lipsum}\usepackage{tcolorbox}\tcbuselibrary{most}\tcbset{mybox/.style={ before={\par\vskip3ex\noindent}, after={\par\vskip1.5ex}}}\newtcolorbox[auto counter=example]{example}[1][]{breakable,mybox,notitle after break,lines before break=5,enhanced,colback=white,colframe=white,coltitle=black,fonttitle=\bfseries,arc=0pt,outer arc=0pt,leftrule=0pt,rightrule=0pt,left=0pt,right=0pt,boxsep=0pt,toptitle=.5\baselineskip,bottomtitle=-.1\baselineskip,title=#1,enlarge top by=4ex,overlay unbroken={ \node[fill=white,anchor=west,font=\bfseries,inner xsep=0pt] at (frame.north west) (title) {Example~\thetcbcounter\enspace}; \draw[thick] (title.east|-frame.north east) -- (frame.north east); \draw[thick] (frame.south west) -- (frame.south east); },overlay first={ \node[fill=white,anchor=west,font=\bfseries,inner xsep=0pt] at (frame.north west) (title) {Example~\thetcbcounter\enspace}; \draw[thick] (title.east|-frame.north east) -- (frame.north east); },overlay last={ \draw[thick] (frame.south west) -- (frame.south east); },}\begin{document}\lipsum[5]\begin{example}[My example]\lipsum[5]\end{example}\lipsum[5]\begin{example}\lipsum[1-3]\end{example}\lipsum[5]\end{document}