1) These sorts of things are actually being studied nowadays with trapped atomic gases. These gases are released from optical traps and expand into what is essentially a perfect vacuum. If you want, they can be recaptured, although a typical trap does not have sharp ``walls'', but harmonic confinement.
2) Yes, if you expand into a vacuum then you just convert internal energy into (collective) kinetic energy $\frac{1}{2}\rho u^2$, where $\rho$ is the mass density and $u$ is the fluid velocity. If you expand forever then temperature will go to zero, and all energy is kinetic.
3) If you recapture then the result will depend somewhat on the nature of the gas and the walls.
i) Ideal gas, soft walls (harmonic confinement). The systems rebounds from the wall and starts to undergo periodic oscillations. The energy remains kinetic. Since the gas is ideal, the motion is undamped.
ii) Real gas, soft walls. Same thing, but collective oscillations are damped by viscosity, Systems settles down, energy goes back to internal (that means for ideal gas equation of state $T$ goes back to initial value), entropy has increased.
iii) Ideal gas, sharp walls. Rebound creates shock waves. This is a complicated process, since you will get a messy systems of interacting shocks. Even ideal shocks create entropy, so this systems may settle down as in ii).
iv) Real gas, sharp wall. As in iii), but this definitely settles down as in ii).
In a adiabatic process, there is no energy supplied to the system. So, when the gas expands adiabatically it is doing work, and the energy to do work can only come from it's internal energy which is why the temperature decreases.
If temperature remains constant, then it is an isothermal process. Since the temperature here is constant,the energy required to do the work must necessarily come from an external agency. Hence, an isothermal process cannot also be an adiabatic process
Best Answer
Pulling the piston does work on the particles. If we consider the pressure of gas as a result of particles hitting the container and being reflected back, then being scattered from a moving piston means that they scatter back with a higher or lower velocity (depending on whether the gas is being compressed or expanding.)
After some energy has been transferred to the gas, it comes to equilibrium via inter-molecular/atomic collisions (although in case of a quasistatic process, the process is so slow that gas always remains in equilibrium.)
The following might be a good image for illustrating the collision between a molecule (B) and a much heavier piston (A):
Related: Proof of pressure of ideal gas from first principles