New/update favicon per page

You're on the right track with using the link rel's on every page, but the trick to forcing the favicon refresh is to append a number at the end of the icon file as a parameter that's always changing (time is the safest bet).

Example with php:

<?php
    echo '<link rel="shortcut icon" href="http://www.yoursite.com/favicon.ico?t=' . time() . '" />';
?>

Doing this makes the browser think the file is new every time it sees it, thus avoiding the caching issue. I hope this points you in the right direction!

You can have a different favicon per page. You'll just need to add the following code to each HTML file:

<link rel="shortcut icon" href="/YourIcon.ico" type="image/x-icon" />