1) You're right. What are you worried you might have missed? Or, why do you think it's discrete?
2) I'm not sure I understand your argument. What do you mean by "extend it to an infinite set where the new open sets are simply equal to the original set"? Extend what?
A hint on (2): A basis element for the box topology is a product of a nonempty open set from each factor space. To also be a basis element for the product topology, all but finitely many of those open sets need to be the entire space in that coordinate. How can you make those coincide?