Autosave nodes, when editing or creating new ones, in database
I'm unaware of any existing production/beta modules with this functionality.
There's an old blog post available from 2014 that discusses the use of GarlicJS to save the current working version of the Drupal node you're working on in browser local storage:
https://www.shooflydesign.org/buzz/lightweight-autosave-for-drupal-with-garlicjs
Quote:

Garlic uses HTML5 local storage, supported by all browsers these days, to cache what you're writing as you write it. I like this solution because, unlike WordPress, it doesn't create a bajillion post revisions. The values are stored by your browser until your submit the form, at which point they're cleared out.

If you're open to updating to Drupal8 (likely not, I realize), then there's a new module available called AutoSave_Form that claims to do something similar to what you're asking for:
https://www.drupal.org/project/autosave_form
Quote:

The autosave form module comes with a configuration form, which could be found under Admin-&gt;Configuration-&gt;Content-&gt;Autosave Form. There the interval could be easily adjusted and decide whether autosave is turned on for content or config entity forms or for both. By default it is active only on content entity forms and currently it is being tested only with content entity forms. If support for config entity forms is required please feel free to open an issue if you find some bugs regarding config entity forms.
At the moment the autosave works only for entity forms, but it will be extended to cover regular forms as well.