I am trying to get the footnotes to reset on each page. I have tried everything I can think of to get it to work, placing "counter-reset: page;" in all of the relevant places. I even tried using "counter-reset: @page;" and a variety of other things to get them to reset. Any suggestions?

Try this:

@page {
    counter-reset: footnote
}

Nope, that did not work. Here are the relevant code pieces with extraneous styles removed:

@page {
	counter-reset: footnote;
}
note {
	display: prince-footnote;
	counter-increment: footnote;
}
note::footnote-call {
	content: "[" attr(n) "]" counter(footnote, lower-alpha);
}
note::footnote-marker {
	footnote-style-position: inside;
	content: "[" attr(n) "]" counter(footnote, lower-alpha);
}

being applied to this tag:

<note type="level4" n="4">footnote text</note>

Sorry, I forgot that this is a new feature that is not available in Prince 5.1. If you like, you can try the latest alpha version of Prince 6.0, which supports resetting the footnote counter on each page and also automatically increments the footnote counter after each footnote. (It has also has a bunch of other new features and improvements).