I would say no. The purpose of sweat is to cool you down. When it evaporates off your skin, it removes heat with it. But since pool water is always going to be cooler than your body, theres no need....unless u have a sweating disorder

Omg do we! Swimming takes so much out of you! its an activity (if done for exercise and not recreation) that is way more effective than jogging and works almost every muscle in your body! Sometimes you will notice if you particularly push yourself that you get hotter, this is when you are sweating. You just cant tell because you dont smell it (sweat only stinks because of bacteria which the chlorine kills dead lol) and you cant see it (its like how you cant tell you are drooling underwater but when you come up everyone else can see that nasty string hangin of your chin)

You do sweat while swimming the water just washes the sweat off of you.

Yes. You most definitely do.