If the process is endothermic that means it gains thermos; right? Yes that means it needs the proper amount of energy to proceed with the reaction.
Why in this case does a gas cool while it expands? What is the difference? I believe the gas expansion you're referring to is the endothermic vaporization of a liquid, which is not the same as the expansion of a gas in terms of volume from gas --> gas. If I'm right in assuming you are talking about this, then these are actually 2 completely seperate reactions as one concerns a phase change while the other only results in a change in volume. Regarding the expansion of gas in terms of volume, the application of this can be seen in the blowing of hot air from your mouth. Air in your mouth is blown through your small mouth opening and as a result, it "feels" cold. But what is happening is that the gas is just expanding and thus resulting in a "cold" temperature.
I am confused; is it the gas that is cooled or its surroundings or both?
If your talking about liquid water evaporating then the gas would have to have a higher temperature than the liquid right? That should make sense. Also the surroundings, might be a little vague to answer your question. If your referring to something like evaporation of water form the skin, then assuming the skin to the surrounding then yes, the surrounding is cooled in the sense that heat is being drawn from it as the water molecules are shifting into the gas phase.
Also, this might be better posted in the Q/A forum. (More people there are chronic checkers...like myself...can never read too much on question/answers lol) For future notes!