Aircraft carriers are gigantic and are designed to be stable. I’ve never been on the largest ones, but as a Marine I was on the Essex (a smaller carrier for jump jets and helicopters) and we sailed basically directly through a typhoon. The ship was pretty stable, pitching about 10 degrees from side to side. Enough to make some experienced sailors toss their lunch, and even if there was no wind, I think it would have been very treacherous to do flight ops with the deck pitching like that.

For fixed wing jets that need to land on deck… even if you ignored the wind that goes along with these conditions, I think recovering aircraft would be too great a risk. That does bring up the wind and visibility that (almost always) goes along with these extreme sea states, which would be probably just as hazardous as the deck pitching about to flight ops.

For basically everything else, I think the naval systems work just fine in bad weather.

In regards to rogue waves, an 85 foot wave would put green water on the flight deck of the carrier I was on, something that would be difficult to imagine for me. Aircraft on the flight deck are chained down with multiple chains, each of the chains having the strength to hold the entire aircraft down, so I can’t imagine losing aircraft, but flooding the planes with sea water probably wouldn’t be good for them.