Why don't monopolists try to establish the highest price possible, as many people allege? What would happen to sales? What about profits?

Monopolists do not charge the highest possible price because charging the highest price doesn't mean earning the highest profit possible. If prices are extremely high, demand will be very low. Low demand means low revenue. Low revenues that come at high costs will lead to low profits, especially if the fixed costs are high. Low production due to low demand means that labor, an opportunity cost, is not used efficiently.