If you already have WoW#1 loaded on Computer 1, you can log into your Battle.net account on Computer 2. It will disconnect you from Battle.net, but it won’t from WoW.
Yes, the license that moves doesn’t keep any of the account bound items on the source account. They’d be able to keep any of their characters though and the progress that they made.
Buying another account would work too. I wasn’t sure if the account your child uses was exclusive to them, or you use it to.
No, it’s actually preferred. Under our policies you can allow one minor, that you are the parent or legal guardian of, access to your account, but having their own makes it a good deal easier all around so that you can control access, chat and purchase restrictions for them without restricting yourself.
If they are under 13 the account should be registered in a parent’s/legal guardian’s name, if they are over 13 but under 18, they should have it in their name with parental controls.
Let me know if you have any other questions, or if I didn’t explain something well enough. 