Shared iCloud Account and Photo Stream, but no longer want to share iCloud accounts
There doesn't seem to be a way to retain the Photo Stream functionality –automatic sharing of every photo taken or saved in both your devices– once you stop sharing the same Apple ID.

You can enable Family Sharing and iCloud Photo Sharing, then manually share the pictures you choose to the "Family" album, which is created in the process. 

You can achieve the same functionality –just the pictures being shared– with iCloud Photo Sharing alone and a shared album.

This question and answers may be of help: How can I share Photos across multiple users on Apple Photos, without paying for iCloud?

Backing up to the same computer it's a little more convoluted, and it's described in extenso in this article at the support site on Apple; the routes described are:


Create multiple user accounts on the computer that will hold the backups.
Sync different playlists to different devices (not sure this is helpful in this situation).
Create a separate iTunes library for each device.


You will want to exercise extra caution on choosing what to sync, to avoid mixing or overwriting each other information (I'm thinking on the options to sync contacts, calendars, email, etc.).

After separating the two accounts, you can regain access to bought music and apps if you set up Family Sharing.