To suppress the parentheses, you can (re)define the footline beamer template (as defined in the outer theme infolines which is internally used by Madrid); for your other question, you can use the optional argument of \title:
\documentclass{beamer}
% Add to preamble from here ------------------
\usetheme[secheader]{Madrid}
\makeatletter
\setbeamertemplate{footline}
{
\leavevmode%
\hbox{%
\begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,center]{author in head/foot}%
\usebeamerfont{author in head/foot}\insertshortauthor%~~\beamer@ifempty{\insertshortinstitute}{}{(\insertshortinstitute)}
\end{beamercolorbox}%
\begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,center]{title in head/foot}%
\usebeamerfont{title in head/foot}\insertshorttitle
\end{beamercolorbox}%
\begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,right]{date in head/foot}%
\usebeamerfont{date in head/foot}\insertshortdate{}\hspace*{2em}
\insertframenumber{} / \inserttotalframenumber\hspace*{2ex}
\end{beamercolorbox}}%
\vskip0pt%
}
\makeatother
% To here ----------------------
\author{A.U. Thor}
\title[The Title in footline]{The Title in the document}
\institute{The Institute}
\begin{document}
\begin{frame}\maketitle\end{frame}
\section{Test section}
\subsection{Test subsection one}
\begin{frame}\frametitle{adafdg}Test\end{frame}
\subsection{Test subsection two}
\begin{frame}Test\end{frame}
\end{document}
EDIT: This is the whole code for a LaTeX user, so just include the marked part in your LyX preamble