For most apps, the only difference is that the Free version has ads. For Calculator HD Pro and Calculator Pro Elite, the paid versions also have 2 exclusive features available as In App Purchases (Widget and Multiple History Tapes).
No, you only have to pay once and then you can download it again for free onto all of your devices.
The app is universal and works with both the iPhone and the iPad.
You can also share your purchase with others in your family if you have Family Sharing enabled on your account.
For more information about Downloading Previous Purchases and Family Sharing, please visit Apple at https://support.apple.com/billing
The Paid version of the app is a totally separate app, so look for a new icon on your device without the word Lite on it.
Start using the Paid version from now on, and delete the Free version from your device so that you do not open it by mistake.
From within the app, go to Settings, and scroll down to the bottom of the first page and tap Restore Purchases.
The Widget can be accessed in your Today View or by 3D Touch on the app icon. For more information, please visit Apple at https://support.apple.com/en-us/HT207122
To see the in app purchase in action and how to use it, watch this video: https://www.youtube.com/watch?v=0I8NdWyPXP0