The simply answer is yes. Steel wool does not have a coating that will protect it from rusting in water. But rusting will take some time.
there is more oxygen in water
Steel wool corrodes from water because the water has oxygen. If the steel wool is in contact with both the water and oxygen it will begin to rust rapidly. The rusting will take a while. It will take between 5-7 days.
the wool does not rust
Steel wool does not get softer under water. That is just your perception.
The steel wool gets wet. If you take the steel wool out (into the air) again it will rust.
Steel wool is not soluble in water, neither can it absorb water but can get wet so if you take the steel wool out of the water the wet (due to capillary action and surface tension) steel wool will weigh more than dry steel wool. Then the steel wool will rust (and the mass will increase because oxygen unites with iron to form the rust).
Depends on whether you leave the steel wool in the Pepsi. If you take it out, the water in the Pepsi will cause the steel wool to rust away. If you leave it in, the phosphoric acid will slowly dissolve the steel wool.
No, gypsum does not corrode stainless steel.
No, because they are plated.
Short answer - yes. Even stainless steel can rust in fresh water, given enough time and chemistry of the water.
yes
They will both rust at the same time.