How did Admiral King survive his early mistakes in World War II?

What good reasons could lead to fire admiral King? Let's look on the different events of the beginning of the war (from the Americain point of view), that were important enough to lead to the firing of Admiral King, had he behaved bad during these events.

There are two types of events: "Bad" ones, defeat for which King could be considered responsible, and "Good" ones, victory in which other American leaders could have felt badly supported by King.

• Bad: Atlantic battle: The organization of convoys was not made partly because of misunderstanding by the USA, partly because they thought that their air patrol capabilities could do enough job before the ships "dilute" in the ocean, and partly because they needed destroyers in the Pacific. What is important is that there were no "big scandal" with a massive consciousness that the 1942 first semester came from a mistake: most Americans military leaders thought they were correct, and in the second semester, US still engaged its forces in Torch operation and Guadalcanal rather than in the battle of the Atlantic
• Bad: Pearl Harbour: not perceived as the mistake of one person
• Good: Coral Sea: success of his subordinates, but he could use it
• Good: Strategic success of King and his subordinates

So overall, by the time of the war, King's achievements were pretty good. There was no reason to replace him.

Admiral King survived what you call mistakes because they were not considered mistakes, but were decisions on how to deal with a situation more involved then the people pointing out the losses during the 'second happy time' acknowledge.

An article discussing this can be seen here (emphasis mine).

In King's defense, noted naval historian Professor Robert W. Love has stated that "Operation Drumbeat (or Paukenschlag) off the Atlantic Coast in early 1942 succeeded largely because the U.S. Navy was already committed to other tasks: transatlantic escort-of-convoy operations, defending troop transports, and maintaining powerful, forward-deployed Atlantic Fleet striking forces to prevent a breakout of heavy German surface forces. Navy leaders, especially Admiral King, were unwilling to risk troop shipping to provide escorts for coastal merchant shipping. Unscheduled, emergency deployments of Army units also created disruptions to navy plans, as did other occasional unexpected tasks. Contrary to the traditional historiography, neither Admiral King's unproven yet widely alleged Anglophobia, an equally undocumented navy reluctance to accept British advice, nor a preference for another strategy caused the delay in the inauguration of costal escort-of-convoy operations. The delay was due to a shortage of escorts, and that resulted from understandably conflicting priorities, a state of affairs that dictated all Allied strategy until 1944."

We might even speculate that that shortage of escorts was due in part to the Destroyers for bases deal, which in late 1940 placed 50 US destroyers into UK hands:

The destroyers-for-bases deal was an agreement between the United States and the United Kingdom on September 2, 1940, according to which 50 Caldwell, Wickes, and Clemson class US Navy destroyers were transferred to the Royal Navy from the US Navy in exchange for land rights on British possessions.

So the US was short of adequate vessels to perform two critical tasks at the same time-protect troop transports heading to England, or protect merchant transports along the US coast. King chose the military option, protecting the troop transports allowing US troops to enter the European Theatre. You can't fight a war if you cant get there. This would be looked at as a tough decision, but King chose protecting the troops over protecting the merchant vessels. Perhaps if we had those 50 destroyers he could have done both. But the first happy time might not have ended without them, so the statistics on these 'happy times' might be quite different.