QUESTION: Does the power of judicial review always require a federal issue in the case? Is it enough that the Supreme Court believes that the state law or state court opinion violates the Constitution (and that in turn creates a federal issue?)
ANSWER: The jurisdiction of the federal courts is constrained by Article III, section 2 of the Constitution. These are all the "cases" or "controversies" that Congress can grant the federal courts the power to decide. The most common "case" is one arising under federal law--a question that concerns a federal law or the federal constitution. So, in a rough sense, yes, cases must generally present a federal question (e.g., this law (state or federal or local or whatever) violates the federal constitution). But some cases that can reach the federal courts do not, the best example being those that come via diversity jurisdiction.
One more point: you have asked specifically about "judicial review." That practice specifically refers to the practice of courts determining whether a law or executive action is constitutional. By definition, if a federal court is reviewing a law's constitutionality, it is addressing a federal question, as it is determining whether the law or executive action is consistent with the federal constitution.
QUESTION: As to Problem 3, I'm confused about the argument made by [the state attorneys general] that the decision NOT to buy insurance is not commercial. I understand the logic behind it, but I don't understand how it applies. If the law states that all persons must buy health insurance, then the decision not to purchase insurance isn't an issue. In fact, it doesn't exist at all. Without an option to not buy insurance, how is the fact that someone who doesn't buy insurance isn't commercial relevant?
ANSWER: I think the issue is logically prior. That is, Congress has enacted a law stating that all Americans must either (a) purchase health insurance by January 1, 2014, or (b) pay a fine. The relevant constitutional question is whether Congress has the authority to regulate behavior in this way. We need to answer this question, regardless of which choice any individual makes. Why is the commerciality of this decision relevant? Because whether the activity being regulated is economic or commercial is the crucial question under the third category according to Lopez and Morrison. So you are right that what a person actually decides is not important to answering the constitutional question. But the nature of the activity being regulated by Congress is critical.