Sure. we can provide a live demo of the app. Just send us a demo request to our support team (email@example.com). Also if required we can configure the app on behalf of you as per your use case.
No. The app does not do any changes to the store theme.
Shopify does not allow individual variants to be hidden. They allow entire product to be hidden. The app hides a product once total inventory (inventory of all the variants of the product) of the product are reached to inventory-threshold defined in the app.
Yes, You can create Hider and Publisher in the app to run at your given time to hide and publish products.
There is a Hide-now button to a Hider, you need to click the hide-now button to cleanup the already sold-out products.