How are the same products with different stock displayed in two separate Warehouses assigned to different Store Views? What status will they have for each Warehouse?

The extension changes the status to "Out of Stock" for a product when its Qty becomes 0. If you want to hide the out-of-stock product on the front-end at all, please navigate to Store >> Catalog >> Inventory and switch "Display Out of Stock Products" option to "No":

By default, this option is set to "Yes".