To answer your questions:

• An individual naval gun was typically referred to by two numbers - one giving the diameter of the bore (and hence of the shell), and one giving the barrel length in multiples of the bore diameter. In common parlance, only the bore diameter is given, so a 16in gun has a bore diameter of 16in.

• Gun capabilities are difficult to measure, but generally speaking, two things are important: the kinetic energy of the shell fired, and the size of the bursting charge in that shell. With a larger gun, a heavier shell is fired, using a greater charge and a longer barrel, meaning that muzzle velocity stays roughly constant. A heavier shell will also retain its velocity for longer, as it is less affected by drag. As the shell is larger, a heavier bursting charge can be carried. For example, the British 12in/45 Mark X fired a ~850 lb AP shell with a muzzle velocity of 2,725 fps. The shell contained a 20lb bursting charge. The 15in/42 Mark I fired a ~1920 lb AP shell at a muzzle velocity of 2,467 fps, with the shell containing a 45lb bursting charge (of a superior explosive). The 16in/45 Mark I fired a 2048 lb shell at ~2,600 fps. This shell had a 50lb bursting charge. However, a smaller gun would tend to have a higher rate of fire.

• In theory, yes. The RN made the decision to use more guns with smaller bores in two cases. The first was the King George V class battleships of 1939. These were designed to fit within the confines of the London Naval Treaty, which placed a maximum limit on the size of guns that could be fitted on new designs. While most nations ignored this, building ships with 15in, 16in or 18in guns, the RN stuck with using 14in guns. As originally designed, the KGVs would have mounted 12 of these in three quadruple turrets. This would have given them a comparable broadside throw weight to their contemporaries. However, problems with weight distribution, and the desire to build the most heavily armoured ship possible on a limited displacement meant that they were constructed with ten guns - two quadruples and one double turret. The other case was with the Town-class cruisers of 1936 and their successors. The 8in guns used on heavy cruisers had the advantage in terms of range and armour penetration over the 6in guns used by light cruisers, though the 6in fired faster. However, a 6in treble turret weighed about as much as an 8in double turret, meaning that on comparably sized ships, the one armed with 6in guns would have an advantage in terms of throw weight. The RN chose to build cruisers armed with 6in guns, as the 6in gun could reach the limits at which fire could effectively be controlled, and because cruisers were not expected to fight ships with heavy armour. This meant that the advantages of the 8in gun were negated, making the 6in gun the much better choice.

• No, armour does not correlate well with gun size, for a number of reasons. Firstly, ships were not uniformly armoured. When a ship is described as having 16in of armour, that refers to the thickest part of the armour, typically the armour belt amidships. Other areas of the ship, especially the ends and superstructure were poorly armoured, or even unarmoured. Damage to these could effectively mission-kill a ship, or cause dangerous flooding. Secondly, range has a great impact on how effectively shells penetrate armour. At close ranges, even the 12in/45 could penetrate armour up to 16in thick. Thirdly, the layout of the armour has a massive impact on how effective it is. The Japanese Yamato had a monolithic armour belt of 16.1in, inclined at 20^o to increase effective thickness. The American South Dakotas had a 12.2in belt inclined at 19^o (plus a 0.8in steel bulkhead), with a 1.25in decapping plate a few feet in front. Going by thickness alone, the Yamatos belt would be more effective, being about 2.5in thicker. However, the South Dakotas was actually more effective - in theory, if both ships were using the most effective possible steel in their armours, the South Dakota would be completely protected against the 15in gun of Bismarck at ranges over 15,300 yards, while Yamato would be protected at ranges over 16,100 yards. This difference was down to the decapping plate, which was intended to strip the armour-penetrating cap from incoming shells, greatly reducing their effectiveness. Added to this is the fact that not all armour is created equally. British armour was very high quality for battleships, as was that used by the French, Italian and German navies. However, the American armour was suited more for the thicknesses used on cruisers and Japanese armour was a generation out of date - with the steel actually used, Yamato was protected against Bismarck's 15in gun at ranges over 21,000 yards. Finally, armour penetration is just as much about the shell as it is about the armour. Poorly designed or made shells can fail to penetrate armour even for large calibres. British shells of WWI were notorious for this - a 6in plate could protect against the 15in shells used at Jutland, while the 13.5in shells in use for the same battle could struggle against 0.5in plates at high obliquities.

• While there was always the theoretical possibility of avoiding action, this was not always practical. Fleets have to achieve objectives, just as land armies do. As a result, battleships and battlefleets could force an action by putting themselves in a position to interfere with their opponents objectives. Jutland occurred because the British fleet put itself sufficiently close to the path of the German fleet, which had been heading north to interfere with British trade through the Skagerrak. The Battle of Cape Spartivento and the Second Battle of Sirte occurred when the Italian Navy attempted to strike against British convoys to Malta, while the Battle of North Cape occurred when the German Scharnhorst attempted to engage a convoy to Russia. Similarly, the Battle of Denmark Strait and the sinking of Bismarck occurred as the result of a German attempt to raid the Atlantic convoy routes. In the Pacific, the Second Naval Battle of Guadalcanal occurred when the Japanese fleet was attempting to bombard Henderson Field on Guadalcanal. Leyte Gulf was a Japanese attempt to disrupt American landings in the Philippines. Battleships could easily destroy any smaller ship, and so they were needed for such times, when a battle could not be avoided.