The second part of your statement I quoted above is how. Recovering is identical except for the declutching (instead, if it is a power generated skid the reaction should be to reduce power). But the true part of the recovery (be it with a manual or automatic tranny) is performing the correct wheel turn in the correct direction and amount to recover. And getting that down requires enough practice so that the reaction becomes muscle memory habitual. Once that is done, recovery in an automatic is just as trivial.
You'd be amazed how many people, who never had any training and have never practiced skid recovery, have the initial reaction of turning the wheel the wrong direction (hint: almost everyone who has not ever practiced the maneuver will do this) - which of course simply makes the skid worse.
I've lived in southern states where people refuse to go faster than 40 if the roads are anything less than bone-dry.
Totally depends on what people are used to driving in.
Take your foot off the gas. If you're still sliding, steer into the slide and correct. If necessary, rinse & repeat.
IMO, the only real advantage a manual has is the ability to do some engine braking.
That's counter-intuitive for many people as typically you want to actually go right.
If you have a manual transmission and you know what you're doing, in some situations you can downshift and use the engine to slow you down more. But in my experience, it's a "nice to have" vs a must for passenger vehicles. (different story for big trucks)
(edit: depending on how far you've already rotated, but I'm speaking as if the front of the car is still ahead of the back of the car)
My car has several modes including Sand and Snow. Both modes will remap the transmission and provide more/less torque. In some modes it will even start in 2 so you don't spin your wheels and get stuck. The throttle response in the pedal will also be increase/decreased as well as the sensitivity of the steering wheel.
Also you have the option to start in 2nd gear to prevent spinning tires and sliding when starting from a dead stop.