2- You can use the restatable environment provided by thmtools.
1- Since the environment generates an anchor, you can use the standard \label, \ref (\pageref) mechanism for cross-referencing.
A complete example:
\documentclass{article}
\usepackage{amsthm}
\usepackage{thmtools, thm-restate}
\usepackage{hyperref}
\declaretheorem{theorem}
\begin{document}
\section{Test Section}
\begin{theorem}
Test theorem
\end{theorem}
\begin{restatable}[Fundamental Theorem of Algebra]{theorem}{fta}
\label{thm:FTA}
Every non-constant single-variable polynomial with complex coefficients has at least one complex root.
\end{restatable}
\appendix
\section[Proof of the Fundamental Theorem of Algebra]{Proof of Theorem~\ref{thm:FTA} (See page~\pageref{thm:FTA})}
\fta*
\begin{proof}
Assume...
\end{proof}
\end{document}

Now that the question has been edited, and the used document class is known, here's a solution using the vldb document class (the link to the class can be found in the question):
\documentclass{vldb}
\usepackage{textcase}
\usepackage{thmtools,thm-restate}
\usepackage{hyperref}
\makeatletter
\def@sect#1#2#3#4#5#6[#7]#8{%
\ifnum #2>\c@secnumdepth
\let@svsec@empty
\else
\refstepcounter{#1}%
\edef@svsec{%
\begingroup
%\ifnum#2>2 \noexpand\rm \fi % changed to next 29 July 2002 gkmt
\ifnum#2>2 \noexpand#6 \fi
\csname the#1\endcsname
\endgroup
\ifnum #2=1\relax .\fi
\hskip 1em
}%
\fi
@tempskipa #5\relax
\ifdim @tempskipa>\z@
\begingroup
#6\relax
@hangfrom{\hskip #3\relax@svsec}%
\begingroup
\interlinepenalty @M
\if@uchead
\MakeTextUppercase{#8}% HERE
\else
#8%
\fi
\par
\endgroup
\endgroup
\csname #1mark\endcsname{#7}%
\vskip -12pt %gkmt, 11 aug 99 and GM July 2000 (was -14) - numbered section head spacing
\addcontentsline{toc}{#1}{%
\ifnum #2>\c@secnumdepth \else
\protect\numberline{\csname the#1\endcsname}%
\fi
#7%
}%
\else
\def@svsechd{%
#6%
\hskip #3\relax
@svsec
\if@uchead
\uppercase{#8}%
\else
#8%
\fi
\csname #1mark\endcsname{#7}%
\addcontentsline{toc}{#1}{%
\ifnum #2>\c@secnumdepth \else
\protect\numberline{\csname the#1\endcsname}%
\fi
#7%
}%
}%
\fi
@xsect{#5}\hskip 1pt
\par
}
\makeatother
\declaretheorem{theorem}
\begin{document}
\section{Test Section}
\begin{theorem}
Test theorem
\end{theorem}
\begin{restatable}[Fundamental Theorem of Algebra]{theorem}{fta}
\label{thm:FTA}
Every non-constant single-variable polynomial with complex coefficients has at least one complex root.
\end{restatable}
\appendix
\section[Proof of the Fundamental Theorem of Algebra]{Proof of Theorem~\ref{thm:FTA}}
\fta*
\begin{proof}
Assume...
\end{proof}
\end{document}

Explanation:
The problem is that vldb.cls sets the section (and also part) titles in upper case and uses the old TeX \uppercase command to this end; this will produce problems if special commands, such as \ref, are inside the argument of \section (or \part) since \uppercase don't recognize them as special commands and tretas them in the wrong way (the LaTeX command \MakeUppercase will also behave bad in this regard). To correct this situation, one can redefine (or patch) the internal command \@sect, to use \MakeTextUppercase (from the textcase package), instead of the problematic \uppercase command. In the example code above, the change was signaled with % HERE.