I'm having trouble creating an Apple ID for my child

Have you setup a “family” with yourself as the “organizer”?

You’ll need to do that in order to create a child’s account.

See:

Create an Apple ID for your child - Apple Support

and

Set up Family Sharing - Apple Support

Make sure the credit card details are entered in your account before trying to enter them in the child's account.

If necessary, contact Apple ID Support for assistance.  See the Apple Support articles below for contact information.  

Apple ID - Official Apple Support.   -

Contact Apple for help with Apple ID account security - Apple Support.  *   (Link doesn’t always load)  

Get Support      -    Work through the screens to get a chat option. You may have to pick a slightly different problem.

Apple Support schedule phone call.   -

Apple on Twitter       

From the Apple Support article below. 

Without proof of your identity via security questions and other carefully selected criteria, Apple Support can't help you reset a password or perform any other actions on your account. These policies are audited and reviewed on a regular basis.

Security and your Apple ID - Apple Support.      *