First off, this is the best place on the internet for information about the Pacific Theater of WWII; and specifically this page is for you. Also try here; I'm going to pull a lot of statistics from there without siting them.
What advantages do extremely large guns give?
The range advantage of larger guns is not that significant. First off, range is determined by a whole slew of factors, and smaller guns can have longer ranges than larger ones. For example, the French 380mm/45 gun had longer range than both the Iowa's 16"/50 and the Yamato's 46cm/45. The German Scharnhorst and Gneisenau had a 28cm gun (equivalent to 11") that had better range than the American 16"/45 gun. Scharnhorst, in fact, with its 11" guns scored the longest range naval gun hit of all time, at 26465 yards, on Glorious, a British aircraft carrier.
So, having demonstrated that range isn't improved that much by larger size, the only advantage remaining is penetration power. This is somewhat significant, because light cruiser's 6" guns were generally unable to damage the belt or tower armor of a 40,000 ton+ battleship. The larger calibers could always pack a larger shell. A larger shell can penetrate a given armor thickness at more angles. Armor that shrugs off a glancing blow from a 12" shell might buckle from the same hit from a 14" shell.
However, once you were in the 14" plus range, the quality of the shell itself became a more significant concern. Shells that large can penetrate pretty much any armor and do massive damage, provided they explode. For example, at the Second Naval Battle of Guadalcanal, South Dakota took 26 large caliber hits from Kirishima, but many of those hits did not explode. Thus South Dakota floated on, and only lost 39 of her crew. Meanwhile, Washington opened fire on Kirishima (at 9000 yards, in case you were wondering, which is a typical engagement range on ships for WWII big guns) and hit her 9 times with large caliber. This was enough to capsize and sink Kirishima within about 4 hours. Crucially, several of Washington's rounds fell short, but hit Kirishima underwater and still exploded. That is a good, reliable shell! The moral of the story is that hitting someone is great, hitting with a round that actually explodes is much better.
What disadvantages do they have?
As you can see in my 'pro' answer, I am not sanguine about larger guns. You should only make a gun large enough to be able to penetrate the enemy's armor at all angles. After that you should concentrate on shell reliability, range-finding equipment (preferably radar...definitely invent that), armor, and damage control.
The disadvantages of large guns are in what they cost you in terms of ship design. Guns have to be mounted topside (obviously) to do any good. Also, you need a lot of them. There are no automatic loaders with the specified technology level, so you are limited by the speed at which you can have men reload them. And since you need (vulnerable) men to reload them, you need the guns to be heavily armored to prevent the men, and ammunition, from being subject to return fire. Turrets were easier to put out of action by killing the operators than by destroying the machinery. The ammunition required for the big guns represents a huge vulnerability that can exploited for the WWI or WWII version of a one-shot.
It is the armor protection for the turret that makes a large gun a liability. If the turret is not armored well enough, you can take out the turret too easily, or potentially doom the ship. The larger the gun, the larger (and heavier) the turret armor. The fact that this weight is topside is significant. You can't let the center of mass rise too high for a number of reasons (not least, the higher the center of mass, the more you heel during a turn, and the less likely you can successfully aim at or hit a target while turning).
If you mount large guns with heavy turrets, high above the water line, you ship must get more massive to compensate. Sure this allows you to add even more armor, but at the expense of even more maneuverability. And this maneuverability is the key weakness. By WWII, the torpedo was a very advanced and effective weapon, despite being unguided. Destroyers and light cruiser could launch spreads of 4 or 6 torpedoes at a time. The true effectiveness of torpedoes in main fleet engagements, battleship to battleship was not really tested in WWII.
The last true fleet battle (until space battles!?!?!) was at Jutland in WWI. Torpedoes were largely inconsequential in that battle, though a storm of torpedoes from destroyers and light cruisers may have allowed the German battleline to escape destruction. However, by WWII torpedoes meant business. At the Battle of Savo Island, the Japanese crushed the American forces, sinking 3 heavy cruisers, by opening with a torpedo salvo at about 12,000 yards. At Tassafaronga, the Japanese were ambushed by the Americans, who found them with radar in the dark. However, the Japanese Long Lance torpedo turned the battle; the Americans did not notice the torpedo launch and did not evade, causing the loss or damage of 4 heavy cruisers. The Americans got their revenge at Vella Gulf and Cape St. George, once they had perfected aiming and firing torpedoes at night guided by radar. Between the two battles, American destroyers sank 6 ships without taking a single hit in return.
Since there was no true battleship to battleship fleet engagement of WWII, I have to conclude that the danger of torpedoes had become very significant. The penetration power of the torpedoes (recall, Yamato was sunk by air-dropped torpedoes) was such that they could stop capital ships.
While the effective engagement range for naval guns, especially on ships with radar, was longer than for torpedoes, torpedoes were enough of a threat at short to medium range that maneuverability was a concern. The increase in turret weight and corresponding increase in ship mass would have been counteracted by the vulnerability of such ships to torpedo attacks. The optimal battleship would not have been any bigger than the Iowa class, and perhaps even smaller.
Why would a nation use them despite the disadvantages?
Simply put, pride and ego.
What would the largest practical battleship gun be?
Guns in the 14" to 16" range were large enough. In a war with no aircraft, the medium range power of the torpedo would have been amply demonstrated. The ideal gun would have the range, accuracy, and rate of fire to engage torpedo carrying ships (destroyers and light cruisers) long before they could launch torpedoes, and the penetration power to punch through a battleship of similar size.
I believe that maneuverability would have been prohibitively bad for battleships above 50,000 tons, and this lack of maneuverability would have made them too vulnerable to torpedo attacks from small escorts. By limiting ship mass to 50,000 tons, total armor is limited such that guns in the 14" to 16" range are all you need to punch a hole in them.
