According to Encyclopedia Britannica, "Early in World War II the primary instrument for delivering naval combat power became the aircraft carrier. The reason was range: aircraft could deliver a concerted attack at 200 miles or more, whereas battleships could do so only at 20 miles or less."
Britannica continues by explaining that, "The foremost tactical question during the transition in the 1920s and ’30s was whether aircraft could lift enough destruction to supersede the battleship. Into the 1930s skeptics were correct that aircraft could not. But by the end of that decade, engines were carrying adequate payloads, dive-bomber and torpedo-plane designs had matured, carrier arresting gear and associated flight-deck handling facilities were up to their tasks, and proficient strike tactics had been well practiced."
If this was the case, why were so many battleships used in WWII? There were plenty of cruisers and destroyers on the Japanese and American sides. What were they all doing? And why did the Japanese build its famous Yamato-class battleships at this time?
This post should serve as an answer to your topline question, though you've already more or less identified it yourself.
As for your other questions, I'll do my best to answer them in turn.
Why were so many battleships used in WWII?
Many of the battleships used in World War II were older ships that had remained in service for decades. I know more about the U.S. Navy and the Japanese Navy, so we'll limit ourselves to those for the moment (Apologies to the Royal Navy). On December 7, 1941, the U.S. Navy included 14 battleships. Of those 14, there were 12 that were built during the 10s and 20s and remained in service. Each of the battleships at Pearl Harbor was among this group of older designs.
There were two new battleships that had just joined the fleet that year: sister ships North Carolina and Washington. These were the first American "fast" battleships that could keep up with the aircraft carriers. Four more modern South Dakota-class battleships were in varying stages of construction on December 7, and each of them would go on to fight in the conflict. Four Iowa-class battleships were in early stages of construction in late 1941; they would all join the fleet during the war (two more Iowas had construction begin but were canceled before completion). Two American battleships were permanently lost at Pearl Harbor (Arizona and Oklahoma). So by the end of World War II, the U.S. Navy had 22 total battleships, 10 of which were new and 12 of which were modernized designs from 20+ years ago.
It's similar for Japan. Omitting the Asahi (no longer considered a battleship by 1941), the Japanese navy had 10 battleships on December 7, none of which was originally built any later than 1921. Yamato was near completion and Musashi joined the fleet in 1942.
Naval construction for most of the interwar period was governed by the Washington and London Naval Treaties. American naval construction was also impacted significantly by the Great Depression. The North Carolina and South Dakota classes weren't authorized until well into the 1930s, after more than a decade without any battleship construction.
There were plenty of cruisers and destroyers on the Japanese and American sides. What were they all doing?
I'll include the utility of the battleship in this answer for organizational purposes. Two answers to this question.
One, it's overly simplistic to suggest that the battleship was rendered obsolete by naval airpower during World War II. A more accurate phrasing would be something along the lines of "The aircraft carrier replaced the battleship as the decisive element of fleet-on-fleet combat." However, there were still lots of other jobs for battleships to do. An obvious example for the battleship is naval bombardment. Carrier air wings could (and did) participate in bombarding enemy defenses before amphibious invasions, but battleships were still well-suited to the mission, and perhaps even better suited in some instances than aircraft carriers. An aircraft sent to attack a land target would need inspection, maintenance, fuel, and would then fly to its target, drop its payload, return to the aircraft carrier, and redo the whole process. Aircraft lacked modern guided munitions (ie, smart bombs), meaning scores of aircraft might need to be committed to a single target to have any chance of destroying it. A warship, on the other hand, could fire its guns over and over again until the target was knocked out. It's worth pointing out here that American World War II-era battleships bombarded land targets in the Korean War, Vietnam War, and during Desert Storm.
Two: Other ships, such as cruisers and destroyers, had their own tasks that aircraft carriers were poorly suited to, or played supporting roles in carrier operations. Nearly every ship played a role in air defense to protect aircraft carriers and other ships from attacking enemy aircraft. Aircraft carriers received formidable anti-air weapons for protection, but battleships and cruisers were marvelous platforms that could be loaded with weaponry to engage aircraft (hopefully) before they reached the carriers. The battleship's main guns were not used for air defense, but the secondary battery of 5-inch guns was an excellent long-range anti-air weapon, and smaller guns on board (the 20mm and 40mm cannon and even smaller machine guns) were all used to engage enemy aircraft. Cruisers are a bit more complicated, because there was great variety in the design of cruisers at the time. Generally speaking, though, cruisers could also be used for shore bombardment or in air defense. In the early war period, when there were few fast battleships, cruisers were the largest ships escorting American aircraft carriers.
There are plenty of other missions or jobs for other ships. Destroyers were more designed for anti-submarine warfare than battleships or cruisers, but were also used for shore bombardment or air defense. Ships were used as command vessels for generals or admirals, for escorting supply ships, as mobile radar outposts, as scouts, or to conduct missions in areas where it was too dangerous to operate aircraft carriers. Even the modern U.S. Navy operates only 11 full-size aircraft carriers among a fleet of about 300 ships.
Why did the Japanese build its famous Yamato-class battleships at this time?
A number of reasons, but partially because Japanese interwar doctrine was centered around fighting a single large-scale engagement to destroy the American fleet. The Japanese fleet was smaller than the U.S. Navy because of interwar naval treaties (and simply because the Japanese economy was smaller than the United States), so the Japanese sought to develop a qualitative superiority to the U.S. Navy. /u/When_Ducks_Attack has an overview here in response to another question. As part of that, the Yamatos were built to be able to destroy any American battleship it would encounter. Most battleships are built to fight other battleships of comparable size, armor, and armament; the Yamatos were bigger, with heavier armor and larger, longer-range guns to outmatch the American battleships of the time.
Each of your five questions here could be its own question, which makes it difficult to provide a deeper response to any of them individually. This should be a good overview, though. If you have any other questions, let me know and I'll do my best to provide more clarity.