Answer to Question 1
Making an economic profit of zero does not mean that the firm is not making any money. It means that it is covering all its costs, including opportunity costs. This means that all resources employed are earning just as much as they would in their next-best use, and thus that there is no gain from moving them to their next -best use.
Answer to Question 2
False. A monopoly never operates in the inelastic portion of its demand curve. Marginal revenue is negative in this region.