How does heat transfer between molecules happen in deep?
A1: There is no displacement of the wall, forces are in equilibrium with zero net force. There are initiated stronger vibrations on warmer wall side by energetic hot air molecules. And weaker vibrations on colder wall side by energy leaching cold water molecules .
A2a: If the wall initial temperature was the cold water temperature, then yes. Otherwise no, as the mean kinetic energy of wall molecules can generally goes both ways, depending on initial temperatures of air, wall and water, weighted by air and water thermal capacity a conductivity.
A2b: Yes
A3: No, as there is no displacement. Hot air with in average more energetic molecules causes stronger vibrations of wall molecules, which cause faster motion of water molecules.
Imagine the whole scenario in more familiar macro world. 2 boxes of ideally elastic balls in gravityless ISS station, separated by a wall of balls interconnected by elastic strings holding all of them at their place.
Now, balls in one of boxes move faster. What will happen during the time ?

The mean Kinetic energy of molecules is proportional to absolute temperature. Their mean speed is then proportional to the square root of absolute temperature and momentum to the absolute temperature.
Pressure on the walls is proportional to momentum change rate, what is proportional to molecule momentum and the rate of collisions. For gases in constant volume, both is proportional to square root of absolute temperature, so pressure is proportional to absolute temperature of ( ideal ) gas.
But pressure is not essential for molecular mechanism of heat transfer.
Molecules of matter at given temperature do not have the same speed and kinetic energy. During their collisions, it fluctuates all the time. What is constant is their mean value.
For monoatomic ideal gas, the mean kinetic energy of 1 mole of the gas is $E_\mathrm{k}=3/2RT$, where R is the universal gas constant about 8.314 J/K/mol. The speed distribution follows the Maxwell-Boltzmann distribution.
It means that for given kinetic energy and speed of a molecule, we cannot say what is the temperature of matter the molecule belongs.
When 2 different regions of matter have different temperature, they exchange heat, what means molecular kinetic energy.
Molecules from adjacent regions mutually exchange kinetic energy by random way by collisions. As by random collisions, they can both gain or lose the energy.
Molecules from both colder and warmer regions mutually both transfer and accept the energy. If temperatures are equal, their mean kinetic energy (*) is equal and the rates of energy flow are equal in both directions. If one region is warmer, the rate of passing energy is greater than the rate of accepting energy.

(*) It is more complicated, as more exactly, it means the mean kinetic energy per each way the molecule can move, called degree of freedom. But the above is still valid for monoatomic gases.