limits to \oplus not working and double ended arrow equation

There is a larger \bigoplus that acts as operator. Or \oplus can be wrapped in \mathop:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\[\bigoplus_{\substack{\rho\in g,\\\rho \neq \rho_0}}\]       
\[\mathop{\oplus}_{\substack{\rho\in g,\\\rho \neq \rho_0}}\]
\end{document}

This is a job for a graphic tool, for example tikz-cd:

\documentclass{article}
\usepackage{tikz-cd,amsmath,amssymb}

\begin{document}

\[
\begin{tikzcd}[row sep=2pt]
{} & \mathrm{VI} \arrow{r} & \mathrm{X} \\
{} \\
\smash[b]{\bigoplus\limits_{\substack{\rho\in\mathfrak{g}\hat{\,}\\\rho\ne\rho_{0}}}}
\vphantom{\bigoplus} \mathrm{V}
  \arrow{uur} \arrow{ddr} \\
{} & & \mathrm{VIII} \arrow{uuu} \\
{} & \mathrm{VII} \arrow{ur} \arrow{dr} \\
{} & & \mathrm{IX}
\end{tikzcd}
\]

\end{document}

The subscripts are smashed out, because otherwise they would make an arrow start too low. Notice \substack for the two row subscript.

You can make the diagram even more compact with row sep=-4pt

As suggested by Qrrbrbirlbel, the \smash trick can be avoided:

\begin{tikzcd}[row sep=-4pt]
{} & \mathrm{VI} \arrow{r} & \mathrm{X} \\
{} \\
|[text depth=+0pt]|\bigoplus\limits_{\substack{\rho\in\mathfrak{g}\hat{\,}\\\rho\ne\rho_{0}}}
\mathrm{V}
  \arrow{uur} \arrow{ddr} \\
{} & & \mathrm{VIII} \arrow{uuu} \\
{} & \mathrm{VII} \arrow{ur} \arrow{dr} \\
{} & & \mathrm{IX}
\end{tikzcd}