You can use a Raspberry Pi, but:
- Install Arch Linux ARM. Then you will have a modern, up to date operating system which includes up to date versions of calibre.
- Install calibre-git from the AUR. The precompiled distro package includes a bug that prevents successful password authentication from mobile devices.
As for multi-user management, that is upcoming and expected in calibre v3 -- which is still being worked on and is nowhere near ready, and there is no projected finish date yet.
As you suggested, adding books at least can be done through auto-import.
- You can save books once they have been corrected and renamed etc., using Save to Disk on the family member's PC.
- Then copy the saved ebook onto the Raspberry Pi.
- Run a script periodically via cron or systemd, using the calibredb backend to add those books to the library.
...
P.S. Your English is more than sufficient to make yourself understood, and many people don't speak English as their first language so grammatical errors are generally excused. You did fine.
Also, feel free to set your location in your profile details, that way people will automatically know to be extra forgiving of your grammar even without you saying anything.