All firms maximize profits where MR=MC. Because of a downward-sloping demand curve and high barriers to entry, monopolists can charge a price higher than MC. Like other price searchers, monopolists take price from the demand curve (at the quantity where MR=MC).
The other statements are false. A monopoly structure is characterized by a well-defined product for which there are no good substitutes. A monopolist will earn a profit as long as the demand curve lies above the average total cost curve (ATC) at the optimal quantity point. Monopolists want to maximize profits, not price.