We have a plugin that allows us to manage custom post types and we'd like to add AJAX functionality in saving, editing, and deleting posts. I couldn't find similar problems in the internet so I'm wondering if this isn't something easily doable?
Step 1: Add AJAX handler to admin_head for new and existing posts.
Step 2: Hook into the save_post action.
This runs after the post has been saved to the database, so you can save whatever postmeta you need and halt the page render.
The only major caveat I can see is that you will need to refresh nonces on the page somehow via XHR.
For deleting a post, I'm not sure why you would want to use AJAX when it's a simple click of a link that does the job already. Deleting a page in every case will send you to another page, namely, edit.php. This defeats the purpose of using AJAX which is to keep everything asynchronous on one page load.
Hope this helps you out.