Where to set the DOCTYPE? My zen subtheme doctype is currently !DOCTYPE html and I would like to change this. What is the correct place to add the doctype I want to use?

I copied the html.tpl.php file and pasted it into the templates directory of my subtheme thinking that's where it's set, but it's not being picked up...
You're in the correct place (html.tpl.php) to modify the doctype assuming you're using Drupal 7, thats the default theme implementation for altering the basic page structure (documentation). In Drupal 6, you're looking for the page.tpl.php file.