That's a pretty standard high speed turn, its what destroyers are designed to do, high-speed maneuvers while chasing or dodging bad guys, or rushing around to reach their station when a formation changes. The skipper orders "left full rudder" and "port back full, all ahead starboard" on the screws.

When a formation goes from, say, "line ahead" to "bent line screen", the signal flags go up to alert the squadron to the upcoming maneuver, and when the flagship pulls them down, the signal executes. For the next few minutes, its all assholes and elbows and ships are rushing about all over the place, flags flying, in a seemingly chaotic, but carefully planned and choreographed confusion. (If somebody screws up there is usually a collision!). When its all over, the ships simultaneously fall into their new formation, often at some new course and speed. Its a pretty awesome sight from the bridge of a destroyer, and it does make you feel all tingly and proud to be even a small part of it.

Its usually preceded by an ominous call over the 1MC "Now stand by for heavy rolls. All hands clear the weather decks while maneuvering at speed." In this case, according to a reader comment, the side was manned by sailors to dress up the ship and impress civilians riding on the carrier alongside. My skipper would never have done that, its dangerous, you might lose someone over the side. Even below, you have to brace yourself to get around during a turn like that, and loose gear is flying around all over the place.

As for the "flex" question, I don't know enough about ship construction to properly answer you. I do know that on my ship (which was approximately of the same size, mission and performance as the USS Gonzalez in this video), the main deck had an 8 inch neoprene spacer or "hinge" running athwartships at the beam which allowed the main deck plates to flex without buckling in a heavy sea. Whether there were arrangements like this all throughout the hull and superstructure, I don't recall.