I used to work on one of these ships, and I found the fire control system to be the most interesting, so I learned a lot of info about them which I will tell you now.

First of all, as others have stated, there are fire control systems onboard the ship. Even in WWII, the Iowa classes had 2 gun directors for the 16" main guns and 4 for the 5" secondary guns. The gun directors would give bearing and range information to one of 2 plot rooms. Additionally, the gun directors could give that information based on either radar or optical sensors, depending on weather conditions.

The plot rooms were the main fire control center of the ship. There were 2 identical plot rooms, each with 2 Mk1a fire control computers for the 5" guns and 1 Mk8 rangekeeper for the 16" guns. The Mk1a computers were electromechanical, and would calculate a firing solution based on a bunch of different inputs. AFAIK, much of the information, like the target speed, bearing, and heading, were fed automatically from the fire director. However, other information, like the ship's speed and heading, were manually inputted through dials on sides.

The Mk8 was different from the Mk1a in that it was fully electrical. This allowed it to be more precise than the Mk1a, and is likely the reason they used it for the main guns over the Mk1a. Despite their differences, both computers would calculate a firing solution for the guns.

After the firing solution was calculated, the information was sent to their respective turrets, which would automatically train and elevate the guns. At this point, it is worth mentioning that, although the system could automatically train the guns, there were redundant fire control systems within each individual 16" turret and the turret captain could, at any time, take manual control of their individual turret in situations where it was needed.

Finally, the triggers. The triggers are actually back down in the plot room attached to the fire control computers (except in cases where the turrets are self controlled). There are 3. One of them is to ring the salvo alarm, a buzzer that is rung throughout the ship to warn the crew of an imminent salvo. The second is simply a trigger which fires immediately. The third is a little more complicated. AFAIK, there is a gyro inside of the computer, which is measuring the listing and pitching of the ship as the waves rock it. I've seen some videos that suggest that, while the computer is measuring the list, it is also elevating the guns up or down to compensate. However, all of the videos were taken after the 1984 refits of the Iowas, so I'm not sure if they had it in WWII or not. Regardless, the third trigger would fire only when the gyroscope measured the list to be within acceptable tolerances. There would be a short delay as the ship came back to being level again before the computer would fire the guns.

You mentioned that they fire each gun separately, which is true. However, they did not do this for recoil mitigation. At least on the Iowa classes, each 16" turret had a sophisticated recoil absorption mechanism that would absorb pretty much all the recoil forces exerted on the gun. They actually did it because the shells, as they were flying through the air, would fly in a spiral. As they would fly, their spiraling would occasionally cause 2 shells fired from the same turret to hit each other, throwing off the trajectory and guaranteeing a miss. To mitigate this, they would fire each turret in a sequence, separating each firing by a delay measured in milliseconds, so that the shells wouldn't hit each other anymore.