Can I enjoy DJ play by using the unit (XDJ-RX) with rekordbox (iOS/Android), without using rekordbox (Mac/Windows)?

Yes, you can enjoy DJ play by using the rekordbox (iOS/Android) installed on the mobile devices, without using rekordbox (Mac/Windows).
The music files already stored on the mobile devices can be loaded on the unit (XDJ-RX) by using rekordbox (iOS/Android).

It is, however, necessary to update the firmware of the XDJ-RX when using rekordbox (Android).
For details, see the link