How can I list registered devices?

This site will show you all the devices you have registered to a specific Apple ID:
https://supportprofile.apple.com

I strongly suggest your daughter use her own Apple ID (and not yours).

If there are previous purchases that subsequently cannot be downloaded because they were purchased using a different Apple ID, you can rectify that in iTunes by doing Store > Authorize This Computer for the different Apple ID, and using that iTunes to sync the iPhone.

Presumably your daughter has her own iTunes, separate and distinct from yours. If her iTunes has your songs, apps, etc. (or vice versa), then the above will work.

Separate Apple IDs will ensure you don't get each other's text messages, FaceTime, etc.

You can no longer register any Apple device with Apple. Apple has discontinued product registration. All you can do now is setup a support profile.

You can see what products are under your support profile by logging into such online.

If you restore the backup of her previous phone you can go to Settings/iTunes&App Stores and see what account it is logged in to.

You can also go to iTunes on your computer, click on iTunes Store, then your Apple ID. Log in, then click Manage Devices.

You can have a new ID, and have apps and music from multiple Apple IDs on a phone (sticking within the license agreement, of course, that limits shared content to the same household). However, any time you update an app you will have to enter the password for the account that the app was purchased with.