So OP there seems to be a lot of misunderstanding of when the Yamatos enter the picture, and what the US actually knew about them.

FOr this we go back to the mid 1930's. Japan is leaving the League of Nations, and is abandoning the Washington Naval Treaty, and the follow on London Treaty which had halted new Battleship construction since the early 20's. Naval attaches in Japan get word and assume that Japan is building new Battleships, and in general all agree that she will probably use 16in guns and be about 45k tons. Much larger than the 35k ton limit adopted under the treaties for any replacement of older ships, additionally the 16in guns were in excess of the 14in standard adopted(however an Escalator Clause was invoked by the US to arm the new North Carolinas with 16in).

Basically without knowing anything but a vague description the West was much more concerned with Japan's rising militarism,a nd their existing battle line, including the Nagato's arguably the best battleships built before 1930. The NC's for instance as the US Navy's first new battleships were designed to be fast enough to keep up with the assumed speed of the IJN's Kongo class not some hypothetical new class. The follow on South Dakotas were much the same but more compact to both attempt to salvage the treaty limits, and to keep an eye on costs, while upping armor but keeping speed. All of this occurred before a final Yamato design was agreed on in Japan let alone known to the West.

The Iowas then were designed beginning in 1938, after the Naval powers minus Japan agreed on a new treaty size limit of 45k tons, which the Iowas were built at. Along with the 33knt speed to keep up with the fleet carriers of the Lexington,Yorktown, and soon to be Essex class. The NC's and SD's while fast enough to keep up as escort, when compared to the old 21 knot speed of the Standard Types, were still not up to the max speed of the carriers(about 32 knots) and if push came to shove could have been left behind in an emergency.

Some rumors though had begun to leak out about the Yamatos and the possible threat did influence the final USN BB design, the Montana designed basically as a larger North Carolina with heavier armor, she would be essentially the same size as the Yamato and about the same speed, just with the 12 16in guns, and because of her much later design(finished in Spring 1942) she incorporated a much larger AAA suite from the beginning. War priorities however meant that construction was never begun.

Yamato herself then was only beginning acceptance trials in early December 1941, and spent the majority of her time sitting at anchor at Kure, or Truk.

Carriers though had been seen to become more effective all the time before 1941, and nobody would have called them useless. Yamamoto though never really completely abandoned the IJN's dream of a surface engagement to annihilate the USN. His plan for Midway in large part feels like he is creating jobs for the BB's to feel included, while the USN wisely left their old slow BB's out of harms way until their new fast battleships joined the fight off Guadalcanal. So while the IJN demonstrated what massed carriers could do, in many ways it was the USN who took those lessons to heart after 9 months of hard schooling.

So as for directly influencing the designs in the West really you can only say that their rumored construction and size impacted one class that was then never built, but their hypothetical capabilities also colored the size and capabilities of the Iowas. In fact it was not till late 1944 that Intel, and Prisoner interrogations even convinced the USN all together that they were armed with guns larger than 16in(in part because the IJN intentionally called the 18in ammo "Special 16in ammo" and it was not till 1944 that econ flights over Truk even took the first picture of the class for the West.

As for their effectiveness, they were not exactly bad ships, just too big for their own good, to unwieldy, too poorly armed with AA initially(and always hurt for Japans lack of a good dual purpose gun). But her effectiveness was also hampered institutionally, the IJN just wasnt as good at Damage Control, lagged in Radar development for both search and Fire Control while shell effectiveness was about even between the Yamato and Iowa, the Yamato though would be much harder to penetrate for the Iowa. There is little doubt though that one on one in just about any conditions in say 1944 the Iowa would be able to handle the Yamato. Her speed could dictate the range of the engagement, while her much greater accuracy at range(thanks to a much better Radar controlled Fire Control system and more automated system vs Japan's reliance on visual control and many more manual inputs for a firing solution) would likely allow her to hit first, which in many ways is most important. While then staying at long range making it harder to be hit, while having her shells able to stand a good shot at penetrating her foes deck armor.

Would look at: http://www.navweaps.com/index_tech/tech-084.htm

http://www.combinedfleet.com/baddest.htm or http://www.combinedfleet.com/b_heavy.htm a comparison of many WW2 BB designs on a site run by the authors of Shattered Sword(Parshall and Tully) a wonderful resource

And any of Norman Friedman's works on Battleships and the US designs in particular.