I know that prince-shrink-to-fit keeps a page within the page width, but is there a way to keep the contents all within one page, shrinking the font or div margins if necessary?

No, there is no mechanism to scale down height at the moment.

Is there a way to have the page height automatically grow to fit everything onto one very long page?

No; that would be similar to what browsers do, but when printing you generally have a particular paper size in mind. May I ask what it is you are trying to do?

Save PDFs of webpages before they disappear off the net forever. Of course, one could also just save the whole page, but then if future browsers display HTML differently, the layout could be disrupted.

A few possibilities suggest themselves: use Prince and specify a very large page height; save screenshots of each page from a browser; or save a copy of each page and a VMware installation with a legacy browser so that you can revisit the page as it was in 2008.