If a father terminates his parental rights does he still have to pay child support?
It varies. In some cases he may be ordered to continue support, but in most cases the support obligation is also terminated. Many judges are reluctant to terminate parental rights because they often assume (without foundation) that the reason for seeking termination is to avoid a support obligation. This line of reasoning has never been shown to be substantive. Certainly there are cases where this is the motivation, but they're the exception rather than the rule.
Is it necessary to legally terminate child support at age 18, or does it happen automatically?
You'll probably want to contact the Child Support Office in your area (and possibly an attorney) to make sure they know that support will be terminating. In some cases, you may need to have the court confirm or order that support be terminated. Don't count on it ending "automatically"- Child Support Offices rarely pay attention to support orders and will often continue to assess support if not explicitly ordered to stop, regardless of what the support order states.
My ex owns his own business, I know this means he can show whatever he wants for income. How can I be sure he is providing correct information?
You would need to get accurate financial information, via a deposition, private investigator(s), and possibly a forensic accountant. You may also subpeona financial records from the bank to find out the true state of the business in terms of income and expenditures.
The ex-wife has moved to several different states without notifying the court or the father, and her whereabouts are unknown. The child is now is foster care. Will he have to pay the back support?
Quite possibly. Even worse, he may also have to pay the State for costs associated for the foster care, whether or not he knew about it or was offered custody of the child. This is unfair, but this is often the way the courts operate.