Does the Lumia Share app work on all Lumia phones?

No, unfortunately it doesn't. The Lumia 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 Lumia Share application will be supported by all Windows Phone 8 and 8.1 Lumias that have Extras+info setting v3.4 or later.