Lots of fun and interesting things in here about the 16" Mark 7 guns mounted on board four of the best fighting ships the world has ever seen. Some misconceptions though too.
-The death and demise of Battleships has nothing to do with cost and everything to do with aircraft and age. Cost during war time is nothing compared to the need to win whatever engagement it was. Unfortunately for the battleship, they ended before true anti-aircraft capabilities were realized. Keep in mind that as advanced as the Iowa's were in the 1940's and 1950's, their AA suite started with the 5"/38cal dual purpose gun mounts on the sides. They were entirely handled by radar and their 5" shells were capped with VT shells (basically a radio controlled detonator) and set to detonate at a specific distance from the target by the directors. This weapon became obsolete for AA duty between world war 2 and the Korean war when jets started taking over. The speed of the aircraft taxed the training and elevating motors to their limits and the shell flight time was simply too low to be effective. The US had realized that this would be a problem as early as 1940 when the design for the Montana Class Battleship was being drummed up. They ditched the 5"/38 for the 5"/54. The extra 16 calibers of barrel length made for a much more effective AA gun. Those barrels made for the Montana's 5"/54's were eventually put on the Midway class carriers (18 of them at the end of WWII). The mid range AA consisted of a boatload (literally) of 40mm Bofors which was again, obsolete by the end of WWII. The US had already developed the 3"/50 cal Mk33 guns that saw service on the Des Moines class cruisers which had more range, better velocity and more punch than the 40mm Bofors, which for WWII was a FANTASTIC AA gun but by the end was rapidly made obsolete. The short range AA was carried by 20mm Oerlikons. None of those 3 weapon systems were comfortable in jet engagements and by the end of Korea, the reactivated Iowa's basically ditched all but a handful of 40mm's and the 5"/38's.
The problem with this assessment is that WWII/Korea was in an awkward time for missile development and a modern day "battleship" designed in today's terms would not suffer from this problem in the slightest. CWIS is nice but it's the LAST LINE of defense for AA duties.
Age was probably the single biggest reason though as the Iowa's were launched in the early 1940's
-Battleship Accuracy is absolutely fantastic given the intended targets and ranges involved. The longest ranged battleship vs battleship engagements occurred out around the 22km range. There were several times in which battleships engaged targets further out than that. Yamato firing at the USS White Planes achieved straddles and a single damaging near miss at ~32km. The Iowa and New Jersey fired upon the fleeing and maneuvering cruiser Nowalki and straddled her at 35km repeatedly until they were ordered to cease fire. In terms of "longest" achieved hits, around 22km is where they occurred. Scharnhorst and Warspite are frequently credited with those hits. Warspite was a WWI survivor pressed into service in WWII. On the USN standpoint, West Virginia achieved hits in five out of her first six salvo's at the Battle of Surigao Strait at a range of ~21km and the fire on Yamashiro was smothering enough to sink her. When you boil down the variables that effect battleship accuracy and consider that these engagements were happening either optically (Scharnhorst and Warspite) or via primitive Radar (WeeVee) and electro-mechanical fire computers it's VERY impressive. People want to cite the War Study accuracy where an Iowa could only get 3% of hits at a broadside Bismarck sized target at 35,000 yards but actual accuracy testing showed that this was on the LOW end of hits actually achieved during live fire testing.
-Vertical stringing on battleships has more to do with velocity variation from barrel to barrel than bedding or barrel whip!
There is a reason that significant time was invested in retrofitting the fire control with DOPPLER radar units and improving powder burn consistency.
-Using the example of either Pearl Harbor or the Bismarck as the downfall of Battleships is a bit short sighted. Pearl Harbor was in a state of peace time readiness. It showed the worst case scenario of when you catch stationary, lightly crewed or non-prepared crews off guard and for all the hub bub about what happened there, only ONE Battleship was a complete loss (Arizona). Oklahoma was refloated and sank enroute to the states for scrapping due to storm damages and quick repairs to refloat her. Utah was not a battleship at the time of her sinking although technically she was a battleship hull, of dated design. The US Battleship fleet faired pretty well all things considered. Bismarck was by herself when she was sank. She had absolutely no assistance in handling 12 ships by herself. She was also not designed with nearly the AA compliment she should have had and Tirpitz's final form showed a more realistic expectation of the needs of AA defense onboard Bismarck class BB's. The USN took AA to the logical conclusion but it was by no means a perfect example of the death of Battleships "Because Carriers." Even Yamato and Musashi can't be effectively used. About the best examples here would be the massive carrier battles at the beginning of the Pacific campaign which showed just how strong a carrier could be against prepared and defended enemies.
-A Mark 8 AP shell definitely needed to hit the ship to be of any real use. Splash damage from the relatively small bursting charge on the shell would be minimal but direct hits were absolutely devastating. As others have said, Jean Bart being hit by Massachusetts was a good example of the power of the Mark 8 shell. Washington & South Dakota vs Kirishima was less of an example as the older Mark 5 2250lb AP shell would have been just as devastating to Kirishima as the Mark 8 was. At 12600 yards, where the Washington opened up on Kirishima, the Mark 8 had nearly enough penetration to penetrate through the ship entirely if it were not for the fuses in the shells to detonate them!
In a lot of ways the 16"/45cal Mark 6 guns were considered superior by BuOrd. The weapon system was well known, limitations well understood, but more importantly the angle of fall from the slower moving shells meant more reliable deck penetration at all ranges. The USN was obsessed with deck penetrations since the early 1900's. All of the US Standard Battleships (Nevadas, Pennsylvania's, New Mexico's, Tennessee's, and Colorados) were designed with more deck armor than their peers and the armor profile was specifically chosen to maximize protection at long ranges. Contrast this with Royal Navy battleship design or German battleship design that still focused on the close engagements of the North Atlantic. The faults of their designs were shown at Jutland in WWI and confirmed multiple times throughout WWII.
-USN Radar in WWII was accurate enough to track the small 5" shell splashes on water and I suspect that had there been sufficient view of the horizon, could have tracked land hits as well although the primary job of the optical gun directors was to track shell splashes since their job had been made basically obsolete by the time the North Carolina Class Battleships hit service in WWII and DEFINITELY by the time the Iowa's entered service. The USN was playing with radar for fire control as early as the mid 1930's.
-The optical horizon at which an Iowa could effectively see was comparable to it's gun range as a ship's superstructure and masts could be seen well enough to get a range. There were ways to get accurate ranges by optical methods at those ranges but Radar was far more reliable and less prone to operator fatigue. Also the US tied their radars directly into the fire control computers which was a significant advantage. Japan had, by wars end, a radar capable of accurately measuring shell splashes and range, etc but it was never tied into any fire control system.
-Washington never got within 8400 yards of Kirishima on that night. South Dakota sailed, due to an electrical problem, within 5,000 yards of Kirishima but never returned fire. Washington began her engagement at 12800 yards and yes, hit with at least 9 confirmed hits, although 20 is more probable given the damage sustained by Kirishima and what was reported by the crew of Kirishima. NavWeaps has a fantastic article on the damage of Kirishima:
http://www.navweaps.com/index_lundgren/Kirishima_Damage_Analysis.pdf
-Early barrel life of battleship guns was quite low but by the time the Iowa's were reactivated in the 1980's the advancements in powder tech reduced barrel ware to the point where the ships would not feasibly wear out the liners in their service.
-South Dakota and Iowa classes are frighteningly similar. The armor design of the Iowa's was nearly identical to the South Dakota in layout and thicknesses. Only a small number of changes to the armor layout were made. A thicker outer hull (1.5" STS armor plate instead of 1.25" STS armor plate on the SoDaks), slightly thicker front bulkhead on the Missouri and Wisconsin (and Kentucky and Illinois, although these two were never completed). The major design consideration for the Iowa was to achieve 33 knots at 45,000 tons of displacement with 16"/50cal rifles on board. To do this they took the SoDaks and basically lengthened them and added more power to get them moving at 33 knots. So while Iowas were newer, there was A LOT of South Dakota in Iowa.
-Float Planes were definitely used to spot shot fall until Radar took over. Unfortunately for USN Floatplane pilots their job was made obsolete basically by the time the US entered WWII. The USN had been toying with radar since the mid 1930's for fire control and had radar accurate enough to track main battery shell splashes by Pearl Harbor and by the end of the war it was accurate enough to not only track the large caliber shell splashes but the small ones and with enough accuracy to correct each battery's fire. Many captains demanded that their floatplanes be removed as they created substantial fire risk on board and the aviation fuel storage could be used for other purposes like storage of more AA ammo!
-An Iowa Class Battleship, ignoring the cost and manpower requirements to do so, would only have one real threat on the modern battlefield: Submarines. That threat has always existed but the Iowa's torpedo protection system was designed for direct impact torpedoes launched shallow enough to hit the side of the ship, not the modern torps designed to detonate under the keel. Nuclear weapons have a near zero chance of being used on a battlefield because of the high probability of world wide destruction following their use. Modern missiles are designed to punch the thin hulls of modern destroyers, cruisers, and aircraft carriers. Guided bombs from aircraft are a threat but the fall back here is that your battleship's AA also includes an extensive battery of surface to air missiles and your battlegroup consisted of plenty more ships with them too. Getting close enough to use a guided AP bomb would be difficult.
-Sub-munitions (Sabot rounds) would extend the usable range of an Iowa Class to absolutely fantastical ranges with relatively low cost but it'd require more testing than has currently been done. The US did play with Saboted 8" rounds in the 16" guns but gave up on it rather quickly.
-The best a modern missile could hope to achieve against an Iowa class battleship is a mission kill and not a full on sinking of the ship. Keep in mind these ships have 12.1" of belt armor backed with .65" of STS armor steel and there is an 1.5" STS armor steel outer hull over all the critical machinery spaces. The Decks are multiple layers of thick armor. 0.75" of STS weather deck with 4.75" of Class B Armor steel and 1.25" of STS a deck below that and 0.65" of STS splinter protection a deck below the main armored deck. If you think a anti-ship missile designed for modern carriers, cruisers, and destroyers is going to do anything to that much armor you are high as a kite! That's not to say you couldn't kill a lot of people or cause serious damage to the combat systems (radar, directors, etc) but you would not sink the ship. The biggest threat to an Iowa today is a submarine.
-The delay in salvo firing of the main guns wasn't to prevent shells from impacting each other but to prevent the blast from the adjacent barrels from impacting the flight of that barrel's shell and moving it slightly. As we all know, small movements at the barrel make huge impacts on target and blast from adjacent barrels fired simultaneously made some very inaccurate shots. FWIW, this is entirely unnecessary as the North Carolinas, South Dakota's, and Iowa's were all three-gun turrets and could fire, elevate, and load independently of the other barrels. This meant your salvo could have significant delays either built in or caused by crew fatigue to minimize this effect. In fact the Mark 7's gun page on Nav Weaps talks about the 1980's dispersion testing firing single barrels only.
As an unrelated note, does anyone have a source for the picture in the OP? I've been looking long and hard for a number of technical details, including shot cards from accuracy testing, of the Iowa's. I'm also looking for details pertaining to her handling characteristics (most notably the tactical diameter/turning radius at various speeds).