Question 1: You can use any of the fields given by \author, \title, \date, or \institute to place the image in the title page; if none of those fields allow you to achieve the desired placement, you can use the textpos package. The example below uses the \author field to add the image.
Question 2: With the help of the textpos package you can add the logo to the frametitle template using \addtobeamertemplate.
A simple example code:
\documentclass{beamer}
\usetheme{Madrid}
\usecolortheme{beaver}
\usepackage{textpos}
\title{The title}
\author[The author]{\includegraphics[height=1cm,width=2cm]{cat}\\The Author}
\institute[Inst.]{The Institute}
\date{\today}
\begin{document}
\begin{frame}
\maketitle
\end{frame}
\addtobeamertemplate{frametitle}{}{%
\begin{textblock*}{100mm}(.85\textwidth,-1cm)
\includegraphics[height=1cm,width=2cm]{cat}
\end{textblock*}}
\begin{frame}{Motivation}
Now the logo is visible
\end{frame}
\end{document}


As osjerick mentions in a comment, the above solution won't behave correctly if \framesubtitle is used (the image will shift downwards); in this case, a TikZ approach can be used to prevent the movement:
\documentclass{beamer}
\usetheme{Madrid}
\usecolortheme{beaver}
\usepackage{tikz}
\title{The title}
\author[The author]{\includegraphics[height=1cm,width=2cm]{cat}\\The Author}
\institute[Inst.]{The Institute}
\date{\today}
\begin{document}
\begin{frame}
\maketitle
\end{frame}
\addtobeamertemplate{frametitle}{}{%
\begin{tikzpicture}[remember picture,overlay]
\node[anchor=north east,yshift=2pt] at (current page.north east) {\includegraphics[height=0.8cm]{cat}};
\end{tikzpicture}}
\begin{frame}{Motivation}
Now the logo is visible
\end{frame}
\begin{frame}{Motivation}
\framesubtitle{A}
Now the logo is visible
\end{frame}
\end{document}
addtobeamertemplate. Now I've understood it. – Ignasi Sep 09 '11 at 07:20\framesubtitle. – osjerick Oct 23 '12 at 00:12\framesubtitle{Some title}figure position is not the same, it moves down slightly. How I can set the figure fixed at top right? – osjerick Oct 23 '12 at 05:43