Last footer of an Exam be "End of Exam"

You can use

\iflastpage{<text 1>}{<text 2>}

inside \footer. <text 1> will be used in the last page, and <text 2> will be used in all other pages, except the last one; a simple example, now including the settings of the edited question:

\documentclass{exam}
\usepackage[paperheight=12cm]{geometry}% just for the example
\usepackage{lipsum}% just to generate filler text for the example

\pagestyle{headandfoot}
\runningheadrule
\firstpageheader{AP Calculus Practice Exam A1}{ Mao Laoshi }{Senior 2}
\firstpageheadrule
\runningheader{AP Calculus Practice Exam A1}
{Page \thepage\ of \numpages}
{Senior 2}
\footer{}
{\iflastpage{End of exam}{Please go on to the next page\ldots}}
{}

\begin{document}
\lipsum[1-7]
\end{document}

One can also achieve the same result with the command \runningfooter{<text>}{<text>}{<text>}:

\runningfooter{}{\iflastpage{End of exam}{Please go on to the next page\ldots}}{Page \thepage\ of \numpages}

This allows the first page footer to be different from the rest of the pages:

\firstpagefooter{}{}{Page \thepage\ of \numpages}