Here is an idea that might answer both your questions. It uses yshift to move the start and end points of the two paths up/down by 5pt. The result could be further improved by shifting the start/end points along the x axis in order for the paths to really be in contact with the nodes' circles.
A custom TikZ style is used to add a strike-through marking at the middle of the second path. This could be parameterized further to allow moving the marking to any position on the path etc.
I cannot tell whether this is the most elegant solution though.
\documentclass{minimal}
\usepackage{tikz}
\usetikzlibrary{decorations,decorations.markings}
\tikzset{
strike through/.style={
postaction=decorate,
decoration={
markings,
mark=at position 0.5 with {
\draw[-] (-5pt,-5pt) -- (5pt, 5pt);
}
}
}
}
\begin{document}
\begin{tikzpicture}
\node[draw,circle] (foo) at (0,0) { foo };
\node[draw,circle] (bar) at (4,0) { bar };
\draw[>=latex,->] ([yshift= 5pt] foo.east) -- ([yshift= 5pt] bar.west);
\draw[>=latex,<-,strike through] ([yshift=-5pt] foo.east) -- ([yshift=-5pt] bar.west);
\end{tikzpicture}
\end{document}
Here is how it looks like:
