I am thinking to get iCloud to backup my phone as well as my family members' devices, can I subscribe the iCloud and they can use the same subscription to make full backup of their devices? How does it work?

You each get 5 GB of free iCloud storage for everything, including backups, but you can only use your own 5 GB. If you upgrade one account to 200 GB of storage and set up Family Sharing then anybody that is part of the family can share in that 200 GB. Your stuff is still private.

iCloud: Set up Family Sharing to use with iCloud.com https://support.apple.com/HT208147

Share an iCloud storage plan with your family - Apple Support - "With Family Sharing, your family can share a single 200GB or 2TB storage plan" https://support.apple.com/HT208147

Whether or not you need to share a plan depends upon your regular iCloud usage. I have very little in iCloud and with 5 devices backed up to iCloud I still only use 1.2 GB of my 5 free.

Yes.

Here's information of what is backed up. Some of it depends upon if you have other iCloud services turned on. For example, if you have sync Contacts turned on it won't be part of the backup because it will already be in its own spot in iCloud and will go back to the phone from there if you have to restore it. That's one of the reasons why my backups category is so tiny, because other things are in their own spots in my 5GB iCloud storage.

About backups for iPhone, iPad, and iPod touch - general comparison of using iTunes vs. iCloud - https://support.apple.com/HT204136

Change your iCloud feature settings [iOS and Mac] - https://support.apple.com/HT207689

Restore your iPhone, iPad, or iPod touch from a backup - https://support.apple.com/HT204184