Can you create iCloud for child (under 13) not through Family Sharing

My daughter has been using my iPhone 4s for the past couple of months essentially to play games. As I am unable to create an Apple ID for her due to her age and as I still wanted to manage the apps being downloaded etc, I have continued to use the same Apple ID on 'her' phone and my newer iPhone 5s. This has been working fine except now I would like to be able to connect/back up the content on her phone to an iCloud account that is not linked to mine as the storage is almost at capacity. Is it possible to create an iCloud account for her, and, can she then use this iCloud account for FaceTime and FindMyiPhone etc or will my Apple ID still have to be used. Sorry if this is a bit confusing. I have been wrecking my brain trying to get my head around it all but to no avail. (P.S. Can't do Family Sharing option as no credit card in household!)