Is the form progress saved when the user goes to the Next page?

Does this method of allowing users to complete forms at a later date enable the "next button" to save the entry? If so, how does the user know where to find the partially filled out form if they get disconnected from the form (accidentally shut down browser or unexpected computer shut down, etc.)

Would the auto save feature work here?

No, that is not the case.

I believe you have mixed this with the session link where the form progress is saved when the user goes to the next page.
Related Help Article: How-to-Save-Forms-and-Continue-Later

The Edit Link is a different link. It can be included in the Autoresponder Email which is sent only after the user submits the form.

The user then can go to the Edit Link and make changes to the data submitted previously but they will have to click the Submit button in order to save the changes.

The progress is not saved when the "Next" button is clicked on this case.