Nokia Lumia 2520 - Does the Nokia Share app work on all Lumia devices?

No, unfortunately it doesn't. The Nokia Share app requires the Extras+Info setting in the device which is preloaded only on Window Phone 8 and onwards.

The application uses a background service that is included in the Extras+info setting v3.4 for Windows Phone 8. There fore the Nokia Share application will be supported by all Windows Phone 8 Lumias that have Extras+info setting v3.4 or later.