Typically no. The parent to whom the back child support is due would have to sue. The money doesn't actually belong to the child, it belongs to the custodial parent. Though if an adult could prove that they supported themselves as a minor, they could perhaps bring a suit. Interesting thought. OR, if the custodial parent has passed away, perhaps the adult child could sue for the back child support as due to the estate.
Almost certainly NOT. But it does depend on where you live. If you are in the US, then it depends on the state in which the court ordered child support payments. You need to either get to a law library, if you know how to use one, or seek the advice of an attorney. The attorney would be your best option. I have never heard of this situation being allowed. The other parent, however, can sue for back payments that were never received, even after the child has left home. Again, it depends largely on the state, and the Judge that hears the case.
I think you need a lawyer for this one, because there are different ways that you can get back child support. You can't just go to court and say that you want the other parent to pay, you have to be able to prove that your child deserves this money, and why you are filing for it. Hope this helps, good luck to you.