I think my thesis requirements are worse, and I satisfied them all with memoir pagestyles and hooks. It's also entirely possible there are more elegant ways of getting this done. Suggestions welcome.
Page 1 of ToC: page number at bottom-center, like plain, with "Page" right justified above page number column:

\copypagestyle{toc-initial}{plain}
\renewcommand{\aftertoctitle}{%
\par\nobreak\vspace{\onelineskip}\vspace{\onelineskip}
\mbox{}\hfill{\normalfont Page}
\par\nobreak
\addtocontents{toc}{\protect\pagestyle{toc}%
\protect\thispagestyle{toc-initial}}%
}
Pages 2-M of ToC, where M is defined as the last page with chapter entries: page number at top-right, with "Page" right-justified above page number column, and "Chapter" left-justified above the chapter entries:

\makepagestyle{toc}
\makeoddhead{toc}{\chaptername}{}{%
\enlargethispage{-3\onelineskip}\normalfont\thepage\par%
\vspace{\onelineskip}Page}
\makeevenhead{toc}{\chaptername}{}{%
\enlargethispage{-3\onelineskip}\normalfont\thepage\par%
\vspace{\onelineskip}Page}
\makeoddfoot{toc}{}{}{}
\makeevenfoot{toc}{}{}{}
Pages M+1-N of ToC (i.e., pages with nothing but Appendix entries): page number at top-right, with "Page" right-justified above page number column, and "Appendix" left-justified above the appendix entries:

\makepagestyle{toc-app}
\makeoddhead{toc-app}{\appendixname}{}{%
\normalfont\thepage\par%
\vspace{\onelineskip}Page}
\makeevenhead{toc-app}{\appendixname}{}{%
\normalfont\thepage\par%
\vspace{\onelineskip}Page}
\makeoddfoot{toc-app}{}{}{}
\makeevenfoot{toc-app}{}{}{}
Page 1 of LoF, LoT, etc: page number at bottom-center, like plain, with "Page" right justified above page number column:

\copypagestyle{lot-initial}{plain}
\renewcommand{\lotheadstart}{\newpage \chapterheadstart}
\renewcommand{\afterlottitle}{%
\let\numberline\oldnumberline
\par\nobreak\vspace{\onelineskip}\vspace{\onelineskip}
{\normalfont Table}\hfill{\normalfont Page}
\par\nobreak\vspace{\onelineskip}
\addtocontents{lot}{\protect\pagestyle{lot}\protect\thispagestyle{lot-initial}}
}
Pages 2-K of LoF, LoT, etc: page number at top-right, with "Page" right-justified above page number column, and "Table," "Figure," etc. left-justified above the appendix entries:

\makepagestyle{lot}
\makeoddhead{lot}{\tablename}{}{%
\enlargethispage{-3\onelineskip}\normalfont\thepage\par\vspace{\onelineskip}Page}
\makeevenhead{lot}{\tablename}{}{%
\enlargethispage{-3\onelineskip}\normalfont\thepage\par\vspace{\onelineskip}Page}
\makeoddfoot{lot}{}{}{}
\makeevenfoot{lot}{}{}{}
There's a bit more on my development version (automagic \checktocheaders, \fixtocheaders, and a few other things to hide the pagestyle commands from my users), but the above is the core of it.
Right now I'm at the point where I can get "Page" inserted into everything I want...but NOW it also inserts in the bibliography! /o\ I tried making that go away with some TeX \if action, but no dice. :(
– Josh Holbrook Apr 23 '11 at 09:47