In general, yes. Per the rules of civil procedure of many/most states, if your claim arises from the same underlying transaction or occurrence as the suit that was filed against you, it is considered...
You simply name the corporation as a defendant by using its corporate name and place of business. Sometimes the actual corporate name is unknown because it is being operated under a business name....
You sure can. It depends what the trustee did such as a breach of fiduciary duty or acting in violation of any statutory laws that govern fiduciaries or any criminal laws.