How best to preserve mods to theme files?

There is no really good way to do this with themes. Sorry. When you change themes, your hacks/mods have to be re-implemented.

However:

3) The addition of scripts needed to talk to external services (eg, Google Analytics). I’m obviously not counting stuff that’s inserted by plugins, since the plugin typically continues to work after a theme upgrade.

In these cases, you’d be best off moving these additions to a custom plugin or somewhere else outside of the theme itself. For example, Google Analytics requires the addition of a small bit of code into the header. This could be accomplished like so, in PHP:
<?php
function insert_my_google_analytics() {
?>
*insert Google code here*
<?php
} // end function
add_action('wp_head','insert_my_google_analytics');
?>


This would be theme independant, since virtually every theme calls the wp_head() function in the header. These are called “action hooks”, and you can find more about them in the codex.

functions.php would be one place, but the problem there is that functions.php is part of the theme. It’s only theme independant if the theme doesn’t have it’s own functions.php.

my-hacks would be a better place, or write your own plugin to fill in these bits. I have a custom plugin specific to my blog where I put hacks like these.