How do I remove the frog picture from the default paper template?

You can delete the frog.jpg image file using the project menu in the editor, and if you then remove the lines:

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{frog.jpg}
\caption{\label{fig:frog}This is a figure caption.}
\end{figure}

from the code, that will remove it from the document.