As you are probably aware, there are often many more lawsuits in the US that in many other countries. Also a company can be sued for a vast multitude of reasons.
In general, if a corporation is sued it is the lawyers that go to court to defend the corporation, not the owner(s). In fact if you are not a lawyer licensed to practice in a particular court, you would not be able to represent the corporation.
As to who pays for the lawyers, you can get many kinds of insurance and what they will pay for and the amounts they will be willing to pay will be spelled out in the particular policies you purchase.
You state the corporation will be selling nutraceutical products. If you were to market them in a way that was contrary to FDA regulations, or you used the mails to send fraudulent goods for example, then it is quite likely that you might be sued by the federal government and they might attempt to extradite you and compel you to defend yourself in court. The mere fact that you have a corporation will not guarantee that you will be immune for civil liability for acts you commit on behalf of the corporation.