Your State's child support agency can do this.
If you're in the US, no, a child cannot sue his parent for child support (payment for child support is not due to the child).
Yes, you can still be a citizen. The US does not revoke citizenship for people who owe child support.
All US states allow income garnishment for court ordered child support.
No. You only need to pay child support if you have a child. Starting a new business is not classed as having a child though it can be expensive.
In the US, there is no statute of limitations on collecting unpaid child support.
Child support is paid to the child's other parent regardless where the child is.
In the US, yes, you have to continue paying child support.
Some countries have reciprocal agreements with the US about child support - contact your State's child support agency for help with this.
If you're in the US... Pregnancy/giving birth does not emancipate a minor, so yes, you are still responsible for child support (but only for her, not for her child of course--the responsibility for child support for her child belongs to that child's father).