Ok so I checked the settings you told me to and the Out of Stock Threshold is 0, the Stock Quantity is the number in hand and the Allow Backorders is not checked.
Let me see if I can explain the problem a little better. So we have products we get in that are used we only get a few of them in. When we get them in I add to the products Stock Quantity. When customers go on to our website and buy these product the Stock Quantity for that product goes down. When it becomes zero the product should auto change to out of stock but it is not all the time. It will allow the customer to believe the product is in stock till they try to add it to there cart. After they try to add it to the cart it will pop up that it is out of stock. I'm not sure why it is doing this because the Stock Quantity will be 0 but the Stock Status for some reason stay as In Stock till I manually change it to Out of Stock.
I hope that helped explain the problem better. It's really driving my customer crazy.
Here is and example : https://www.outdoordogsupply.com/product/refurbished-garmin-dc50-gps-collar/