What is the "properties" column inside the Diary entries table good for?

As of now, the properties column serves no purpose.

Though, we are looking into adding features like tagging entries as important in the future.
The properties column then will contain the meta data for each entry needed to provide such a feature.

Who has the permission to create, edit or delete entries?

To be able to create, edit and delete entries you need to have page edit permissions.
For people that are unable to edit your page there will be no editor prompt and edit / delete buttons.

Are my entries gone forever if I delete them?

No! This is one of the many great things about storing all your entries as page content.
Should you ever want to recover deleted entries, you can simply go to your page history and revert your page if needed.

How can I change the date format of my Diary?

Changing the date format can be done via the General Configuration tab in your Confluence Administration. There you can change the date format which the diary uses.