The number of question types depends on how specifically you classify them. The number is in the neighborhood of 15 or so, but someone might, for instance, group "must be true" and "most strongly supports" questions into the same category (e.g. "Inference"). Others may define them as falling under different categories (e.g. "Strong Inference" and "Weak Inference").
Whether to use a conditional statement and/or the contrapositive is not always determined by the question type. In main point questions, you generally can ignore conditionals altogether. In parallel questions you almost always have to deal with them, and sometimes with their contrapositives. In other question types, their use may depend on whether there even is a statement of a conditional or causal nature in the argument. Even then, sometimes it's not necessary to worry about it or its contrapositive.