Out of Stock on "Shop" and "Product category" page

Hi,

We have just received our first order through a woocommerce plugin and there were a few issues regarding the “out of stock” message.

1. the item didn’t automatically update to be ‘out of stock’ even when the payment had been received (there was just 1 quantity of this item) – shouldn’t this change automatically?

2. the order did not appear as “processing” automatically – I changed it manually. – Shouldn’t this change automatically?

3. I had to manually change the stock from 1 to 0 – shouldn’t it update automatically when the payment goes through?

4. There is no ‘out of stock’ message on the shop or category pages, just on the product page – is there a way to change this?

Maybe I have set it up incorrectly. If you have any ideas, please share.

Thanks

https://www.ads-software.com/plugins/woocommerce/