When do I formally "Own" an app from App Store?
Once you tap Get, then confirm it by tapping again, then entering your Apple ID or using Touch ID, and the download starts, then you own it.

The download starting is important—if there's not enough space on the device, or the network disconnects first, or anything else prevents the circular progress bar from starting, you don't own the app.

Once download has started, you can tap the Stop button to stop the download. The icon changes to an iCloud download icon and you now own the app.
You can download apps you own from the Purchases list found at the top of the Updates tab on iOS.
Once the app has been added to your Apple ID, the app will remain on the list regardless of whether the app developer changes the price tier of the app. It won't be removed if the price increases and you can't be charged in the future.

This also applies with part 2, as if you purchase the app, then delete it, you can download it again without being charged even if the price tier increases.