I want to turn this off as I have a custom ordering process using custom fields.
You can't turn it off, because it's hard-coded into the submit metabox for the post post type, you can however hide the sticky checkbox and update the sticky array to unstick any currently stuck posts.
Hide the sticky checkbox
Add additional CSS to the post and post-new pages
Unstick any currently stuck posts
Add this one of your theme files, load a page that calls that file, then remove
Hope that helps. :)
Your code was really helpful for me, but the "sticky post" option is still showing on the quick edit at the "All posts" page, so I've made some changes to fix this: