The battleship era was probably over by 1920. Only a few knew it then.
For the most part, yes. There were still a few slugfests in the Pacific. The Hood did not fare so well against the Bismarck, either, and pocket battleships were relevant as well. The second World War was the end of the era, though as the Carriers came into their own. Had the Bismarck and Tirpitz gained the North Atlantic, their ability to wipe out convoys delivering materiel to the European Theater (and the Eastern Front via Murmansk) would have been a game changer. Air power ended that plan.
Strongly disagree, the Battleship did not become entirely obsolete until the early 1960s with the operational deployment of the carrier launched all weather strike fighter. Carrier launched planes with a useful anti-ship payload did not enter service until the mid 1930s, prior to that they were a support arm for the big guns acting as scouts, spotters and protection from the enemy doing the same. As late as the battle of Surigao Straight there were battles between fleets where the air arm played very little role.